Calculus Examples

$y = 4 \tan (2 x) - \sin^{3} (5 x)$

Step 1

Find the first derivative.

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Step 1.1

By the Sum Rule, the derivative of $4 \tan (2 x) - \sin^{3} (5 x)$ with respect to $x$ is $\frac{d}{d x} [4 \tan (2 x)] + \frac{d}{d x} [- \sin^{3} (5 x)]$ .

$\frac{d}{d x} [4 \tan (2 x)] + \frac{d}{d x} [- \sin^{3} (5 x)]$

Step 1.2

Evaluate $\frac{d}{d x} [4 \tan (2 x)]$ .

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Step 1.2.1

Since $4$ is constant with respect to $x$ , the derivative of $4 \tan (2 x)$ with respect to $x$ is $4 \frac{d}{d x} [\tan (2 x)]$ .

$4 \frac{d}{d x} [\tan (2 x)] + \frac{d}{d x} [- \sin^{3} (5 x)]$

Step 1.2.2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \tan (x)$ and $g (x) = 2 x$ .

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Step 1.2.2.1

To apply the Chain Rule, set $u_{1}$ as $2 x$ .

$4 (\frac{d}{d u_{1}} [\tan (u_{1})] \frac{d}{d x} [2 x]) + \frac{d}{d x} [- \sin^{3} (5 x)]$

Step 1.2.2.2

The derivative of $\tan (u_{1})$ with respect to $u_{1}$ is $\sec^{2} (u_{1})$ .

$4 (\sec^{2} (u_{1}) \frac{d}{d x} [2 x]) + \frac{d}{d x} [- \sin^{3} (5 x)]$

Step 1.2.2.3

Replace all occurrences of $u_{1}$ with $2 x$ .

$4 (\sec^{2} (2 x) \frac{d}{d x} [2 x]) + \frac{d}{d x} [- \sin^{3} (5 x)]$

Step 1.2.3

Since $2$ is constant with respect to $x$ , the derivative of $2 x$ with respect to $x$ is $2 \frac{d}{d x} [x]$ .

$4 (\sec^{2} (2 x) (2 \frac{d}{d x} [x])) + \frac{d}{d x} [- \sin^{3} (5 x)]$

Step 1.2.4

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$4 (\sec^{2} (2 x) (2 \cdot 1)) + \frac{d}{d x} [- \sin^{3} (5 x)]$

Step 1.2.5

Multiply $2$ by $1$ .

$4 (\sec^{2} (2 x) \cdot 2) + \frac{d}{d x} [- \sin^{3} (5 x)]$

Step 1.2.6

Move $2$ to the left of $\sec^{2} (2 x)$ .

$4 (2 \cdot \sec^{2} (2 x)) + \frac{d}{d x} [- \sin^{3} (5 x)]$

Step 1.2.7

Multiply $2$ by $4$ .

$8 \sec^{2} (2 x) + \frac{d}{d x} [- \sin^{3} (5 x)]$

Step 1.3

Evaluate $\frac{d}{d x} [- \sin^{3} (5 x)]$ .

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Step 1.3.1

Since $- 1$ is constant with respect to $x$ , the derivative of $- \sin^{3} (5 x)$ with respect to $x$ is $- \frac{d}{d x} [\sin^{3} (5 x)]$ .

$8 \sec^{2} (2 x) - \frac{d}{d x} [\sin^{3} (5 x)]$

Step 1.3.2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{3}$ and $g (x) = \sin (5 x)$ .

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Step 1.3.2.1

To apply the Chain Rule, set $u_{2}$ as $\sin (5 x)$ .

$8 \sec^{2} (2 x) - (\frac{d}{d u_{2}} [{u_{2}}^{3}] \frac{d}{d x} [\sin (5 x)])$

Step 1.3.2.2

Differentiate using the Power Rule which states that $\frac{d}{d u_{2}} [{u_{2}}^{n}]$ is $n {u_{2}}^{n - 1}$ where $n = 3$ .

$8 \sec^{2} (2 x) - (3 {u_{2}}^{2} \frac{d}{d x} [\sin (5 x)])$

Step 1.3.2.3

Replace all occurrences of $u_{2}$ with $\sin (5 x)$ .

$8 \sec^{2} (2 x) - (3 \sin^{2} (5 x) \frac{d}{d x} [\sin (5 x)])$

Step 1.3.3

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \sin (x)$ and $g (x) = 5 x$ .

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Step 1.3.3.1

To apply the Chain Rule, set $u_{3}$ as $5 x$ .

$8 \sec^{2} (2 x) - (3 \sin^{2} (5 x) (\frac{d}{d u_{3}} [\sin (u_{3})] \frac{d}{d x} [5 x]))$

Step 1.3.3.2

The derivative of $\sin (u_{3})$ with respect to $u_{3}$ is $\cos (u_{3})$ .

$8 \sec^{2} (2 x) - (3 \sin^{2} (5 x) (\cos (u_{3}) \frac{d}{d x} [5 x]))$

Step 1.3.3.3

Replace all occurrences of $u_{3}$ with $5 x$ .

$8 \sec^{2} (2 x) - (3 \sin^{2} (5 x) (\cos (5 x) \frac{d}{d x} [5 x]))$

Step 1.3.4

Since $5$ is constant with respect to $x$ , the derivative of $5 x$ with respect to $x$ is $5 \frac{d}{d x} [x]$ .

$8 \sec^{2} (2 x) - (3 \sin^{2} (5 x) (\cos (5 x) (5 \frac{d}{d x} [x])))$

Step 1.3.5

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$8 \sec^{2} (2 x) - (3 \sin^{2} (5 x) (\cos (5 x) (5 \cdot 1)))$

Step 1.3.6

Multiply $5$ by $1$ .

$8 \sec^{2} (2 x) - (3 \sin^{2} (5 x) (\cos (5 x) \cdot 5))$

Step 1.3.7

Move $5$ to the left of $\cos (5 x)$ .

$8 \sec^{2} (2 x) - (3 \sin^{2} (5 x) (5 \cdot \cos (5 x)))$

Step 1.3.8

Multiply $5$ by $3$ .

$8 \sec^{2} (2 x) - (15 \sin^{2} (5 x) \cos (5 x))$

Step 1.3.9

Multiply $15$ by $- 1$ .

$8 \sec^{2} (2 x) - 15 \sin^{2} (5 x) \cos (5 x)$

Step 1.4

Reorder terms.

$f' (x) = - 15 \sin^{2} (5 x) \cos (5 x) + 8 \sec^{2} (2 x)$

Step 2

Find the second derivative.

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Step 2.1

By the Sum Rule, the derivative of $- 15 \sin^{2} (5 x) \cos (5 x) + 8 \sec^{2} (2 x)$ with respect to $x$ is $\frac{d}{d x} [- 15 \sin^{2} (5 x) \cos (5 x)] + \frac{d}{d x} [8 \sec^{2} (2 x)]$ .

$\frac{d}{d x} [- 15 \sin^{2} (5 x) \cos (5 x)] + \frac{d}{d x} [8 \sec^{2} (2 x)]$

Step 2.2

Evaluate $\frac{d}{d x} [- 15 \sin^{2} (5 x) \cos (5 x)]$ .

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Step 2.2.1

Since $- 15$ is constant with respect to $x$ , the derivative of $- 15 \sin^{2} (5 x) \cos (5 x)$ with respect to $x$ is $- 15 \frac{d}{d x} [\sin^{2} (5 x) \cos (5 x)]$ .

$- 15 \frac{d}{d x} [\sin^{2} (5 x) \cos (5 x)] + \frac{d}{d x} [8 \sec^{2} (2 x)]$

Step 2.2.2

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = \sin^{2} (5 x)$ and $g (x) = \cos (5 x)$ .

$- 15 (\sin^{2} (5 x) \frac{d}{d x} [\cos (5 x)] + \cos (5 x) \frac{d}{d x} [\sin^{2} (5 x)]) + \frac{d}{d x} [8 \sec^{2} (2 x)]$

Step 2.2.3

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \cos (x)$ and $g (x) = 5 x$ .

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Step 2.2.3.1

To apply the Chain Rule, set $u_{1}$ as $5 x$ .

$- 15 (\sin^{2} (5 x) (\frac{d}{d u_{1}} [\cos (u_{1})] \frac{d}{d x} [5 x]) + \cos (5 x) \frac{d}{d x} [\sin^{2} (5 x)]) + \frac{d}{d x} [8 \sec^{2} (2 x)]$

Step 2.2.3.2

The derivative of $\cos (u_{1})$ with respect to $u_{1}$ is $- \sin (u_{1})$ .

$- 15 (\sin^{2} (5 x) (- \sin (u_{1}) \frac{d}{d x} [5 x]) + \cos (5 x) \frac{d}{d x} [\sin^{2} (5 x)]) + \frac{d}{d x} [8 \sec^{2} (2 x)]$

Step 2.2.3.3

Replace all occurrences of $u_{1}$ with $5 x$ .

$- 15 (\sin^{2} (5 x) (- \sin (5 x) \frac{d}{d x} [5 x]) + \cos (5 x) \frac{d}{d x} [\sin^{2} (5 x)]) + \frac{d}{d x} [8 \sec^{2} (2 x)]$

Step 2.2.4

Since $5$ is constant with respect to $x$ , the derivative of $5 x$ with respect to $x$ is $5 \frac{d}{d x} [x]$ .

$- 15 (\sin^{2} (5 x) (- \sin (5 x) (5 \frac{d}{d x} [x])) + \cos (5 x) \frac{d}{d x} [\sin^{2} (5 x)]) + \frac{d}{d x} [8 \sec^{2} (2 x)]$

Step 2.2.5

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$- 15 (\sin^{2} (5 x) (- \sin (5 x) (5 \cdot 1)) + \cos (5 x) \frac{d}{d x} [\sin^{2} (5 x)]) + \frac{d}{d x} [8 \sec^{2} (2 x)]$

Step 2.2.6

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{2}$ and $g (x) = \sin (5 x)$ .

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Step 2.2.6.1

To apply the Chain Rule, set $u_{2}$ as $\sin (5 x)$ .

$- 15 (\sin^{2} (5 x) (- \sin (5 x) (5 \cdot 1)) + \cos (5 x) (\frac{d}{d u_{2}} [{u_{2}}^{2}] \frac{d}{d x} [\sin (5 x)])) + \frac{d}{d x} [8 \sec^{2} (2 x)]$

Step 2.2.6.2

Differentiate using the Power Rule which states that $\frac{d}{d u_{2}} [{u_{2}}^{n}]$ is $n {u_{2}}^{n - 1}$ where $n = 2$ .

$- 15 (\sin^{2} (5 x) (- \sin (5 x) (5 \cdot 1)) + \cos (5 x) (2 u_{2} \frac{d}{d x} [\sin (5 x)])) + \frac{d}{d x} [8 \sec^{2} (2 x)]$

Step 2.2.6.3

Replace all occurrences of $u_{2}$ with $\sin (5 x)$ .

$- 15 (\sin^{2} (5 x) (- \sin (5 x) (5 \cdot 1)) + \cos (5 x) (2 \sin (5 x) \frac{d}{d x} [\sin (5 x)])) + \frac{d}{d x} [8 \sec^{2} (2 x)]$

Step 2.2.7

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \sin (x)$ and $g (x) = 5 x$ .

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Step 2.2.7.1

To apply the Chain Rule, set $u_{3}$ as $5 x$ .

$- 15 (\sin^{2} (5 x) (- \sin (5 x) (5 \cdot 1)) + \cos (5 x) (2 \sin (5 x) (\frac{d}{d u_{3}} [\sin (u_{3})] \frac{d}{d x} [5 x]))) + \frac{d}{d x} [8 \sec^{2} (2 x)]$

Step 2.2.7.2

The derivative of $\sin (u_{3})$ with respect to $u_{3}$ is $\cos (u_{3})$ .

$- 15 (\sin^{2} (5 x) (- \sin (5 x) (5 \cdot 1)) + \cos (5 x) (2 \sin (5 x) (\cos (u_{3}) \frac{d}{d x} [5 x]))) + \frac{d}{d x} [8 \sec^{2} (2 x)]$

Step 2.2.7.3

Replace all occurrences of $u_{3}$ with $5 x$ .

$- 15 (\sin^{2} (5 x) (- \sin (5 x) (5 \cdot 1)) + \cos (5 x) (2 \sin (5 x) (\cos (5 x) \frac{d}{d x} [5 x]))) + \frac{d}{d x} [8 \sec^{2} (2 x)]$

Step 2.2.8

Since $5$ is constant with respect to $x$ , the derivative of $5 x$ with respect to $x$ is $5 \frac{d}{d x} [x]$ .

$- 15 (\sin^{2} (5 x) (- \sin (5 x) (5 \cdot 1)) + \cos (5 x) (2 \sin (5 x) (\cos (5 x) (5 \frac{d}{d x} [x])))) + \frac{d}{d x} [8 \sec^{2} (2 x)]$

Step 2.2.9

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$- 15 (\sin^{2} (5 x) (- \sin (5 x) (5 \cdot 1)) + \cos (5 x) (2 \sin (5 x) (\cos (5 x) (5 \cdot 1)))) + \frac{d}{d x} [8 \sec^{2} (2 x)]$

Step 2.2.10

Multiply $5$ by $1$ .

$- 15 (\sin^{2} (5 x) (- \sin (5 x) \cdot 5) + \cos (5 x) (2 \sin (5 x) (\cos (5 x) (5 \cdot 1)))) + \frac{d}{d x} [8 \sec^{2} (2 x)]$

Step 2.2.11

Multiply $5$ by $- 1$ .

$- 15 (\sin^{2} (5 x) (- 5 \sin (5 x)) + \cos (5 x) (2 \sin (5 x) (\cos (5 x) (5 \cdot 1)))) + \frac{d}{d x} [8 \sec^{2} (2 x)]$

Step 2.2.12

Multiply $\sin^{2} (5 x)$ by $\sin (5 x)$ by adding the exponents.

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Step 2.2.12.1

Move $\sin (5 x)$ .

$- 15 (\sin (5 x) \sin^{2} (5 x) \cdot - 5 + \cos (5 x) (2 \sin (5 x) (\cos (5 x) (5 \cdot 1)))) + \frac{d}{d x} [8 \sec^{2} (2 x)]$

Step 2.2.12.2

Multiply $\sin (5 x)$ by $\sin^{2} (5 x)$ .

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Step 2.2.12.2.1

Raise $\sin (5 x)$ to the power of $1$ .

$- 15 (\sin^{1} (5 x) \sin^{2} (5 x) \cdot - 5 + \cos (5 x) (2 \sin (5 x) (\cos (5 x) (5 \cdot 1)))) + \frac{d}{d x} [8 \sec^{2} (2 x)]$

Step 2.2.12.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$- 15 ({\sin (5 x)}^{1 + 2} \cdot - 5 + \cos (5 x) (2 \sin (5 x) (\cos (5 x) (5 \cdot 1)))) + \frac{d}{d x} [8 \sec^{2} (2 x)]$

Step 2.2.12.3

Add $1$ and $2$ .

$- 15 (\sin^{3} (5 x) \cdot - 5 + \cos (5 x) (2 \sin (5 x) (\cos (5 x) (5 \cdot 1)))) + \frac{d}{d x} [8 \sec^{2} (2 x)]$

Step 2.2.13

Move $- 5$ to the left of $\sin^{3} (5 x)$ .

$- 15 (- 5 \cdot \sin^{3} (5 x) + \cos (5 x) (2 \sin (5 x) (\cos (5 x) (5 \cdot 1)))) + \frac{d}{d x} [8 \sec^{2} (2 x)]$

Step 2.2.14

Multiply $5$ by $1$ .

$- 15 (- 5 \sin^{3} (5 x) + \cos (5 x) (2 \sin (5 x) (\cos (5 x) \cdot 5))) + \frac{d}{d x} [8 \sec^{2} (2 x)]$

Step 2.2.15

Move $5$ to the left of $\cos (5 x)$ .

$- 15 (- 5 \sin^{3} (5 x) + \cos (5 x) (2 \sin (5 x) (5 \cdot \cos (5 x)))) + \frac{d}{d x} [8 \sec^{2} (2 x)]$

Step 2.2.16

Multiply $5$ by $2$ .

$- 15 (- 5 \sin^{3} (5 x) + \cos (5 x) (10 \sin (5 x) \cos (5 x))) + \frac{d}{d x} [8 \sec^{2} (2 x)]$

Step 2.2.17

Raise $\cos (5 x)$ to the power of $1$ .

$- 15 (- 5 \sin^{3} (5 x) + 10 \sin (5 x) (\cos^{1} (5 x) \cos (5 x))) + \frac{d}{d x} [8 \sec^{2} (2 x)]$

Step 2.2.18

Raise $\cos (5 x)$ to the power of $1$ .

$- 15 (- 5 \sin^{3} (5 x) + 10 \sin (5 x) (\cos^{1} (5 x) \cos^{1} (5 x))) + \frac{d}{d x} [8 \sec^{2} (2 x)]$

Step 2.2.19

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$- 15 (- 5 \sin^{3} (5 x) + 10 \sin (5 x) {\cos (5 x)}^{1 + 1}) + \frac{d}{d x} [8 \sec^{2} (2 x)]$

Step 2.2.20

Add $1$ and $1$ .

$- 15 (- 5 \sin^{3} (5 x) + 10 \sin (5 x) \cos^{2} (5 x)) + \frac{d}{d x} [8 \sec^{2} (2 x)]$

Step 2.3

Evaluate $\frac{d}{d x} [8 \sec^{2} (2 x)]$ .

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Step 2.3.1

Since $8$ is constant with respect to $x$ , the derivative of $8 \sec^{2} (2 x)$ with respect to $x$ is $8 \frac{d}{d x} [\sec^{2} (2 x)]$ .

$- 15 (- 5 \sin^{3} (5 x) + 10 \sin (5 x) \cos^{2} (5 x)) + 8 \frac{d}{d x} [\sec^{2} (2 x)]$

Step 2.3.2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{2}$ and $g (x) = \sec (2 x)$ .

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Step 2.3.2.1

To apply the Chain Rule, set $u_{4}$ as $\sec (2 x)$ .

$- 15 (- 5 \sin^{3} (5 x) + 10 \sin (5 x) \cos^{2} (5 x)) + 8 (\frac{d}{d u_{4}} [{u_{4}}^{2}] \frac{d}{d x} [\sec (2 x)])$

Step 2.3.2.2

Differentiate using the Power Rule which states that $\frac{d}{d u_{4}} [{u_{4}}^{n}]$ is $n {u_{4}}^{n - 1}$ where $n = 2$ .

$- 15 (- 5 \sin^{3} (5 x) + 10 \sin (5 x) \cos^{2} (5 x)) + 8 (2 u_{4} \frac{d}{d x} [\sec (2 x)])$

Step 2.3.2.3

Replace all occurrences of $u_{4}$ with $\sec (2 x)$ .

$- 15 (- 5 \sin^{3} (5 x) + 10 \sin (5 x) \cos^{2} (5 x)) + 8 (2 \sec (2 x) \frac{d}{d x} [\sec (2 x)])$

Step 2.3.3

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \sec (x)$ and $g (x) = 2 x$ .

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Step 2.3.3.1

To apply the Chain Rule, set $u_{5}$ as $2 x$ .

$- 15 (- 5 \sin^{3} (5 x) + 10 \sin (5 x) \cos^{2} (5 x)) + 8 (2 \sec (2 x) (\frac{d}{d u_{5}} [\sec (u_{5})] \frac{d}{d x} [2 x]))$

Step 2.3.3.2

The derivative of $\sec (u_{5})$ with respect to $u_{5}$ is $\sec (u_{5}) \tan (u_{5})$ .

$- 15 (- 5 \sin^{3} (5 x) + 10 \sin (5 x) \cos^{2} (5 x)) + 8 (2 \sec (2 x) (\sec (u_{5}) \tan (u_{5}) \frac{d}{d x} [2 x]))$

Step 2.3.3.3

Replace all occurrences of $u_{5}$ with $2 x$ .

$- 15 (- 5 \sin^{3} (5 x) + 10 \sin (5 x) \cos^{2} (5 x)) + 8 (2 \sec (2 x) (\sec (2 x) \tan (2 x) \frac{d}{d x} [2 x]))$

Step 2.3.4

Since $2$ is constant with respect to $x$ , the derivative of $2 x$ with respect to $x$ is $2 \frac{d}{d x} [x]$ .

$- 15 (- 5 \sin^{3} (5 x) + 10 \sin (5 x) \cos^{2} (5 x)) + 8 (2 \sec (2 x) (\sec (2 x) \tan (2 x) (2 \frac{d}{d x} [x])))$

Step 2.3.5

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$- 15 (- 5 \sin^{3} (5 x) + 10 \sin (5 x) \cos^{2} (5 x)) + 8 (2 \sec (2 x) (\sec (2 x) \tan (2 x) (2 \cdot 1)))$

Step 2.3.6

Multiply $2$ by $1$ .

$- 15 (- 5 \sin^{3} (5 x) + 10 \sin (5 x) \cos^{2} (5 x)) + 8 (2 \sec (2 x) (\sec (2 x) \tan (2 x) \cdot 2))$

Step 2.3.7

Move $2$ to the left of $\sec (2 x) \tan (2 x)$ .

$- 15 (- 5 \sin^{3} (5 x) + 10 \sin (5 x) \cos^{2} (5 x)) + 8 (2 \sec (2 x) (2 \cdot (\sec (2 x) \tan (2 x))))$

Step 2.3.8

Multiply $2$ by $2$ .

$- 15 (- 5 \sin^{3} (5 x) + 10 \sin (5 x) \cos^{2} (5 x)) + 8 (4 \sec (2 x) (\sec (2 x) \tan (2 x)))$

Step 2.3.9

Raise $\sec (2 x)$ to the power of $1$ .

$- 15 (- 5 \sin^{3} (5 x) + 10 \sin (5 x) \cos^{2} (5 x)) + 8 (4 (\sec^{1} (2 x) \sec (2 x)) \tan (2 x))$

Step 2.3.10

Raise $\sec (2 x)$ to the power of $1$ .

$- 15 (- 5 \sin^{3} (5 x) + 10 \sin (5 x) \cos^{2} (5 x)) + 8 (4 (\sec^{1} (2 x) \sec^{1} (2 x)) \tan (2 x))$

Step 2.3.11

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$- 15 (- 5 \sin^{3} (5 x) + 10 \sin (5 x) \cos^{2} (5 x)) + 8 (4 {\sec (2 x)}^{1 + 1} \tan (2 x))$

Step 2.3.12

Add $1$ and $1$ .

$- 15 (- 5 \sin^{3} (5 x) + 10 \sin (5 x) \cos^{2} (5 x)) + 8 (4 \sec^{2} (2 x) \tan (2 x))$

Step 2.3.13

Multiply $4$ by $8$ .

$- 15 (- 5 \sin^{3} (5 x) + 10 \sin (5 x) \cos^{2} (5 x)) + 32 \sec^{2} (2 x) \tan (2 x)$

Step 2.4

Simplify.

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Step 2.4.1

Apply the distributive property.

$- 15 (- 5 \sin^{3} (5 x)) - 15 (10 \sin (5 x) \cos^{2} (5 x)) + 32 \sec^{2} (2 x) \tan (2 x)$

Step 2.4.2

Combine terms.

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Step 2.4.2.1

Multiply $- 5$ by $- 15$ .

$75 \sin^{3} (5 x) - 15 (10 \sin (5 x) \cos^{2} (5 x)) + 32 \sec^{2} (2 x) \tan (2 x)$

Step 2.4.2.2

Multiply $10$ by $- 15$ .

$75 \sin^{3} (5 x) - 150 \sin (5 x) \cos^{2} (5 x) + 32 \sec^{2} (2 x) \tan (2 x)$

Step 2.4.3

Reorder terms.

$- 150 \cos^{2} (5 x) \sin (5 x) + 32 \sec^{2} (2 x) \tan (2 x) + 75 \sin^{3} (5 x)$

Step 2.4.4

Simplify each term.

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Step 2.4.4.1

Rewrite $\sec (2 x)$ in terms of sines and cosines.

$- 150 \cos^{2} (5 x) \sin (5 x) + 32 {(\frac{1}{\cos (2 x)})}^{2} \tan (2 x) + 75 \sin^{3} (5 x)$

Step 2.4.4.2

Apply the product rule to $\frac{1}{\cos (2 x)}$ .

$- 150 \cos^{2} (5 x) \sin (5 x) + 32 \frac{1^{2}}{\cos^{2} (2 x)} \tan (2 x) + 75 \sin^{3} (5 x)$

Step 2.4.4.3

One to any power is one.

$- 150 \cos^{2} (5 x) \sin (5 x) + 32 \frac{1}{\cos^{2} (2 x)} \tan (2 x) + 75 \sin^{3} (5 x)$

Step 2.4.4.4

Combine $32$ and $\frac{1}{\cos^{2} (2 x)}$ .

$- 150 \cos^{2} (5 x) \sin (5 x) + \frac{32}{\cos^{2} (2 x)} \tan (2 x) + 75 \sin^{3} (5 x)$

Step 2.4.4.5

Rewrite $\tan (2 x)$ in terms of sines and cosines.

$- 150 \cos^{2} (5 x) \sin (5 x) + \frac{32}{\cos^{2} (2 x)} \cdot \frac{\sin (2 x)}{\cos (2 x)} + 75 \sin^{3} (5 x)$

Step 2.4.4.6

Combine.

$- 150 \cos^{2} (5 x) \sin (5 x) + \frac{32 \sin (2 x)}{\cos^{2} (2 x) \cos (2 x)} + 75 \sin^{3} (5 x)$

Step 2.4.4.7

Multiply $\cos^{2} (2 x)$ by $\cos (2 x)$ by adding the exponents.

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Step 2.4.4.7.1

Multiply $\cos^{2} (2 x)$ by $\cos (2 x)$ .

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Step 2.4.4.7.1.1

Raise $\cos (2 x)$ to the power of $1$ .

$- 150 \cos^{2} (5 x) \sin (5 x) + \frac{32 \sin (2 x)}{\cos^{2} (2 x) \cos^{1} (2 x)} + 75 \sin^{3} (5 x)$

Step 2.4.4.7.1.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$- 150 \cos^{2} (5 x) \sin (5 x) + \frac{32 \sin (2 x)}{{\cos (2 x)}^{2 + 1}} + 75 \sin^{3} (5 x)$

Step 2.4.4.7.2

Add $2$ and $1$ .

$- 150 \cos^{2} (5 x) \sin (5 x) + \frac{32 \sin (2 x)}{\cos^{3} (2 x)} + 75 \sin^{3} (5 x)$

Step 2.4.5

Simplify each term.

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Step 2.4.5.1

Factor $\cos (2 x)$ out of $\cos^{3} (2 x)$ .

$- 150 \cos^{2} (5 x) \sin (5 x) + \frac{32 \sin (2 x)}{\cos (2 x) \cos^{2} (2 x)} + 75 \sin^{3} (5 x)$

Step 2.4.5.2

Separate fractions.

$- 150 \cos^{2} (5 x) \sin (5 x) + \frac{32}{\cos^{2} (2 x)} \cdot \frac{\sin (2 x)}{\cos (2 x)} + 75 \sin^{3} (5 x)$

Step 2.4.5.3

Convert from $\frac{\sin (2 x)}{\cos (2 x)}$ to $\tan (2 x)$ .

$- 150 \cos^{2} (5 x) \sin (5 x) + \frac{32}{\cos^{2} (2 x)} \tan (2 x) + 75 \sin^{3} (5 x)$

Step 2.4.5.4

Multiply by $1$ .

$- 150 \cos^{2} (5 x) \sin (5 x) + \frac{32}{\cos^{2} (2 x) \cdot 1} \tan (2 x) + 75 \sin^{3} (5 x)$

Step 2.4.5.5

Separate fractions.

$- 150 \cos^{2} (5 x) \sin (5 x) + \frac{32}{1} \cdot \frac{1}{\cos^{2} (2 x)} \tan (2 x) + 75 \sin^{3} (5 x)$

Step 2.4.5.6

Convert from $\frac{1}{\cos^{2} (2 x)}$ to $\sec^{2} (2 x)$ .

$- 150 \cos^{2} (5 x) \sin (5 x) + \frac{32}{1} \sec^{2} (2 x) \tan (2 x) + 75 \sin^{3} (5 x)$

Step 2.4.5.7

Divide $32$ by $1$ .

$f'' (x) = - 150 \cos^{2} (5 x) \sin (5 x) + 32 \sec^{2} (2 x) \tan (2 x) + 75 \sin^{3} (5 x)$

Step 3

Find the third derivative.

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Step 3.1

By the Sum Rule, the derivative of $- 150 \cos^{2} (5 x) \sin (5 x) + 32 \sec^{2} (2 x) \tan (2 x) + 75 \sin^{3} (5 x)$ with respect to $x$ is $\frac{d}{d x} [- 150 \cos^{2} (5 x) \sin (5 x)] + \frac{d}{d x} [32 \sec^{2} (2 x) \tan (2 x)] + \frac{d}{d x} [75 \sin^{3} (5 x)]$ .

$\frac{d}{d x} [- 150 \cos^{2} (5 x) \sin (5 x)] + \frac{d}{d x} [32 \sec^{2} (2 x) \tan (2 x)] + \frac{d}{d x} [75 \sin^{3} (5 x)]$

Step 3.2

Evaluate $\frac{d}{d x} [- 150 \cos^{2} (5 x) \sin (5 x)]$ .

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Step 3.2.1

Since $- 150$ is constant with respect to $x$ , the derivative of $- 150 \cos^{2} (5 x) \sin (5 x)$ with respect to $x$ is $- 150 \frac{d}{d x} [\cos^{2} (5 x) \sin (5 x)]$ .

$- 150 \frac{d}{d x} [\cos^{2} (5 x) \sin (5 x)] + \frac{d}{d x} [32 \sec^{2} (2 x) \tan (2 x)] + \frac{d}{d x} [75 \sin^{3} (5 x)]$

Step 3.2.2

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = \cos^{2} (5 x)$ and $g (x) = \sin (5 x)$ .

$- 150 (\cos^{2} (5 x) \frac{d}{d x} [\sin (5 x)] + \sin (5 x) \frac{d}{d x} [\cos^{2} (5 x)]) + \frac{d}{d x} [32 \sec^{2} (2 x) \tan (2 x)] + \frac{d}{d x} [75 \sin^{3} (5 x)]$

Step 3.2.3

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \sin (x)$ and $g (x) = 5 x$ .

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Step 3.2.3.1

To apply the Chain Rule, set $u_{1}$ as $5 x$ .

$- 150 (\cos^{2} (5 x) (\frac{d}{d u_{1}} [\sin (u_{1})] \frac{d}{d x} [5 x]) + \sin (5 x) \frac{d}{d x} [\cos^{2} (5 x)]) + \frac{d}{d x} [32 \sec^{2} (2 x) \tan (2 x)] + \frac{d}{d x} [75 \sin^{3} (5 x)]$

Step 3.2.3.2

The derivative of $\sin (u_{1})$ with respect to $u_{1}$ is $\cos (u_{1})$ .

$- 150 (\cos^{2} (5 x) (\cos (u_{1}) \frac{d}{d x} [5 x]) + \sin (5 x) \frac{d}{d x} [\cos^{2} (5 x)]) + \frac{d}{d x} [32 \sec^{2} (2 x) \tan (2 x)] + \frac{d}{d x} [75 \sin^{3} (5 x)]$

Step 3.2.3.3

Replace all occurrences of $u_{1}$ with $5 x$ .

$- 150 (\cos^{2} (5 x) (\cos (5 x) \frac{d}{d x} [5 x]) + \sin (5 x) \frac{d}{d x} [\cos^{2} (5 x)]) + \frac{d}{d x} [32 \sec^{2} (2 x) \tan (2 x)] + \frac{d}{d x} [75 \sin^{3} (5 x)]$

Step 3.2.4

Since $5$ is constant with respect to $x$ , the derivative of $5 x$ with respect to $x$ is $5 \frac{d}{d x} [x]$ .

$- 150 (\cos^{2} (5 x) (\cos (5 x) (5 \frac{d}{d x} [x])) + \sin (5 x) \frac{d}{d x} [\cos^{2} (5 x)]) + \frac{d}{d x} [32 \sec^{2} (2 x) \tan (2 x)] + \frac{d}{d x} [75 \sin^{3} (5 x)]$

Step 3.2.5

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$- 150 (\cos^{2} (5 x) (\cos (5 x) (5 \cdot 1)) + \sin (5 x) \frac{d}{d x} [\cos^{2} (5 x)]) + \frac{d}{d x} [32 \sec^{2} (2 x) \tan (2 x)] + \frac{d}{d x} [75 \sin^{3} (5 x)]$

Step 3.2.6

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{2}$ and $g (x) = \cos (5 x)$ .

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Step 3.2.6.1

To apply the Chain Rule, set $u_{2}$ as $\cos (5 x)$ .

$- 150 (\cos^{2} (5 x) (\cos (5 x) (5 \cdot 1)) + \sin (5 x) (\frac{d}{d u_{2}} [{u_{2}}^{2}] \frac{d}{d x} [\cos (5 x)])) + \frac{d}{d x} [32 \sec^{2} (2 x) \tan (2 x)] + \frac{d}{d x} [75 \sin^{3} (5 x)]$

Step 3.2.6.2

Differentiate using the Power Rule which states that $\frac{d}{d u_{2}} [{u_{2}}^{n}]$ is $n {u_{2}}^{n - 1}$ where $n = 2$ .

$- 150 (\cos^{2} (5 x) (\cos (5 x) (5 \cdot 1)) + \sin (5 x) (2 u_{2} \frac{d}{d x} [\cos (5 x)])) + \frac{d}{d x} [32 \sec^{2} (2 x) \tan (2 x)] + \frac{d}{d x} [75 \sin^{3} (5 x)]$

Step 3.2.6.3

Replace all occurrences of $u_{2}$ with $\cos (5 x)$ .

$- 150 (\cos^{2} (5 x) (\cos (5 x) (5 \cdot 1)) + \sin (5 x) (2 \cos (5 x) \frac{d}{d x} [\cos (5 x)])) + \frac{d}{d x} [32 \sec^{2} (2 x) \tan (2 x)] + \frac{d}{d x} [75 \sin^{3} (5 x)]$

Step 3.2.7

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \cos (x)$ and $g (x) = 5 x$ .

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Step 3.2.7.1

To apply the Chain Rule, set $u_{3}$ as $5 x$ .

$- 150 (\cos^{2} (5 x) (\cos (5 x) (5 \cdot 1)) + \sin (5 x) (2 \cos (5 x) (\frac{d}{d u_{3}} [\cos (u_{3})] \frac{d}{d x} [5 x]))) + \frac{d}{d x} [32 \sec^{2} (2 x) \tan (2 x)] + \frac{d}{d x} [75 \sin^{3} (5 x)]$

Step 3.2.7.2

The derivative of $\cos (u_{3})$ with respect to $u_{3}$ is $- \sin (u_{3})$ .

$- 150 (\cos^{2} (5 x) (\cos (5 x) (5 \cdot 1)) + \sin (5 x) (2 \cos (5 x) (- \sin (u_{3}) \frac{d}{d x} [5 x]))) + \frac{d}{d x} [32 \sec^{2} (2 x) \tan (2 x)] + \frac{d}{d x} [75 \sin^{3} (5 x)]$

Step 3.2.7.3

Replace all occurrences of $u_{3}$ with $5 x$ .

$- 150 (\cos^{2} (5 x) (\cos (5 x) (5 \cdot 1)) + \sin (5 x) (2 \cos (5 x) (- \sin (5 x) \frac{d}{d x} [5 x]))) + \frac{d}{d x} [32 \sec^{2} (2 x) \tan (2 x)] + \frac{d}{d x} [75 \sin^{3} (5 x)]$

Step 3.2.8

Since $5$ is constant with respect to $x$ , the derivative of $5 x$ with respect to $x$ is $5 \frac{d}{d x} [x]$ .

$- 150 (\cos^{2} (5 x) (\cos (5 x) (5 \cdot 1)) + \sin (5 x) (2 \cos (5 x) (- \sin (5 x) (5 \frac{d}{d x} [x])))) + \frac{d}{d x} [32 \sec^{2} (2 x) \tan (2 x)] + \frac{d}{d x} [75 \sin^{3} (5 x)]$

Step 3.2.9

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$- 150 (\cos^{2} (5 x) (\cos (5 x) (5 \cdot 1)) + \sin (5 x) (2 \cos (5 x) (- \sin (5 x) (5 \cdot 1)))) + \frac{d}{d x} [32 \sec^{2} (2 x) \tan (2 x)] + \frac{d}{d x} [75 \sin^{3} (5 x)]$

Step 3.2.10

Multiply $5$ by $1$ .

$- 150 (\cos^{2} (5 x) (\cos (5 x) \cdot 5) + \sin (5 x) (2 \cos (5 x) (- \sin (5 x) (5 \cdot 1)))) + \frac{d}{d x} [32 \sec^{2} (2 x) \tan (2 x)] + \frac{d}{d x} [75 \sin^{3} (5 x)]$

Step 3.2.11

Move $5$ to the left of $\cos (5 x)$ .

$- 150 (\cos^{2} (5 x) (5 \cdot \cos (5 x)) + \sin (5 x) (2 \cos (5 x) (- \sin (5 x) (5 \cdot 1)))) + \frac{d}{d x} [32 \sec^{2} (2 x) \tan (2 x)] + \frac{d}{d x} [75 \sin^{3} (5 x)]$

Step 3.2.12

Multiply $\cos^{2} (5 x)$ by $\cos (5 x)$ by adding the exponents.

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Step 3.2.12.1

Move $\cos (5 x)$ .

$- 150 (\cos (5 x) \cos^{2} (5 x) \cdot 5 + \sin (5 x) (2 \cos (5 x) (- \sin (5 x) (5 \cdot 1)))) + \frac{d}{d x} [32 \sec^{2} (2 x) \tan (2 x)] + \frac{d}{d x} [75 \sin^{3} (5 x)]$

Step 3.2.12.2

Multiply $\cos (5 x)$ by $\cos^{2} (5 x)$ .

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Step 3.2.12.2.1

Raise $\cos (5 x)$ to the power of $1$ .

$- 150 (\cos^{1} (5 x) \cos^{2} (5 x) \cdot 5 + \sin (5 x) (2 \cos (5 x) (- \sin (5 x) (5 \cdot 1)))) + \frac{d}{d x} [32 \sec^{2} (2 x) \tan (2 x)] + \frac{d}{d x} [75 \sin^{3} (5 x)]$

Step 3.2.12.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$- 150 ({\cos (5 x)}^{1 + 2} \cdot 5 + \sin (5 x) (2 \cos (5 x) (- \sin (5 x) (5 \cdot 1)))) + \frac{d}{d x} [32 \sec^{2} (2 x) \tan (2 x)] + \frac{d}{d x} [75 \sin^{3} (5 x)]$

Step 3.2.12.3

Add $1$ and $2$ .

$- 150 (\cos^{3} (5 x) \cdot 5 + \sin (5 x) (2 \cos (5 x) (- \sin (5 x) (5 \cdot 1)))) + \frac{d}{d x} [32 \sec^{2} (2 x) \tan (2 x)] + \frac{d}{d x} [75 \sin^{3} (5 x)]$

Step 3.2.13

Move $5$ to the left of $\cos^{3} (5 x)$ .

$- 150 (5 \cdot \cos^{3} (5 x) + \sin (5 x) (2 \cos (5 x) (- \sin (5 x) (5 \cdot 1)))) + \frac{d}{d x} [32 \sec^{2} (2 x) \tan (2 x)] + \frac{d}{d x} [75 \sin^{3} (5 x)]$

Step 3.2.14

Multiply $5$ by $1$ .

$- 150 (5 \cos^{3} (5 x) + \sin (5 x) (2 \cos (5 x) (- \sin (5 x) \cdot 5))) + \frac{d}{d x} [32 \sec^{2} (2 x) \tan (2 x)] + \frac{d}{d x} [75 \sin^{3} (5 x)]$

Step 3.2.15

Multiply $5$ by $- 1$ .

$- 150 (5 \cos^{3} (5 x) + \sin (5 x) (2 \cos (5 x) (- 5 \sin (5 x)))) + \frac{d}{d x} [32 \sec^{2} (2 x) \tan (2 x)] + \frac{d}{d x} [75 \sin^{3} (5 x)]$

Step 3.2.16

Multiply $- 5$ by $2$ .

$- 150 (5 \cos^{3} (5 x) + \sin (5 x) (- 10 \cos (5 x) \sin (5 x))) + \frac{d}{d x} [32 \sec^{2} (2 x) \tan (2 x)] + \frac{d}{d x} [75 \sin^{3} (5 x)]$

Step 3.2.17

Raise $\sin (5 x)$ to the power of $1$ .

$- 150 (5 \cos^{3} (5 x) - 10 \cos (5 x) (\sin^{1} (5 x) \sin (5 x))) + \frac{d}{d x} [32 \sec^{2} (2 x) \tan (2 x)] + \frac{d}{d x} [75 \sin^{3} (5 x)]$

Step 3.2.18

Raise $\sin (5 x)$ to the power of $1$ .

$- 150 (5 \cos^{3} (5 x) - 10 \cos (5 x) (\sin^{1} (5 x) \sin^{1} (5 x))) + \frac{d}{d x} [32 \sec^{2} (2 x) \tan (2 x)] + \frac{d}{d x} [75 \sin^{3} (5 x)]$

Step 3.2.19

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$- 150 (5 \cos^{3} (5 x) - 10 \cos (5 x) {\sin (5 x)}^{1 + 1}) + \frac{d}{d x} [32 \sec^{2} (2 x) \tan (2 x)] + \frac{d}{d x} [75 \sin^{3} (5 x)]$

Step 3.2.20

Add $1$ and $1$ .

$- 150 (5 \cos^{3} (5 x) - 10 \cos (5 x) \sin^{2} (5 x)) + \frac{d}{d x} [32 \sec^{2} (2 x) \tan (2 x)] + \frac{d}{d x} [75 \sin^{3} (5 x)]$

Step 3.3

Evaluate $\frac{d}{d x} [32 \sec^{2} (2 x) \tan (2 x)]$ .

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Step 3.3.1

Since $32$ is constant with respect to $x$ , the derivative of $32 \sec^{2} (2 x) \tan (2 x)$ with respect to $x$ is $32 \frac{d}{d x} [\sec^{2} (2 x) \tan (2 x)]$ .

$- 150 (5 \cos^{3} (5 x) - 10 \cos (5 x) \sin^{2} (5 x)) + 32 \frac{d}{d x} [\sec^{2} (2 x) \tan (2 x)] + \frac{d}{d x} [75 \sin^{3} (5 x)]$

Step 3.3.2

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = \sec^{2} (2 x)$ and $g (x) = \tan (2 x)$ .

$- 150 (5 \cos^{3} (5 x) - 10 \cos (5 x) \sin^{2} (5 x)) + 32 (\sec^{2} (2 x) \frac{d}{d x} [\tan (2 x)] + \tan (2 x) \frac{d}{d x} [\sec^{2} (2 x)]) + \frac{d}{d x} [75 \sin^{3} (5 x)]$

Step 3.3.3

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \tan (x)$ and $g (x) = 2 x$ .

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Step 3.3.3.1

To apply the Chain Rule, set $u_{4}$ as $2 x$ .

$- 150 (5 \cos^{3} (5 x) - 10 \cos (5 x) \sin^{2} (5 x)) + 32 (\sec^{2} (2 x) (\frac{d}{d u_{4}} [\tan (u_{4})] \frac{d}{d x} [2 x]) + \tan (2 x) \frac{d}{d x} [\sec^{2} (2 x)]) + \frac{d}{d x} [75 \sin^{3} (5 x)]$

Step 3.3.3.2

The derivative of $\tan (u_{4})$ with respect to $u_{4}$ is $\sec^{2} (u_{4})$ .

$- 150 (5 \cos^{3} (5 x) - 10 \cos (5 x) \sin^{2} (5 x)) + 32 (\sec^{2} (2 x) (\sec^{2} (u_{4}) \frac{d}{d x} [2 x]) + \tan (2 x) \frac{d}{d x} [\sec^{2} (2 x)]) + \frac{d}{d x} [75 \sin^{3} (5 x)]$

Step 3.3.3.3

Replace all occurrences of $u_{4}$ with $2 x$ .

$- 150 (5 \cos^{3} (5 x) - 10 \cos (5 x) \sin^{2} (5 x)) + 32 (\sec^{2} (2 x) (\sec^{2} (2 x) \frac{d}{d x} [2 x]) + \tan (2 x) \frac{d}{d x} [\sec^{2} (2 x)]) + \frac{d}{d x} [75 \sin^{3} (5 x)]$

Step 3.3.4

Since $2$ is constant with respect to $x$ , the derivative of $2 x$ with respect to $x$ is $2 \frac{d}{d x} [x]$ .

$- 150 (5 \cos^{3} (5 x) - 10 \cos (5 x) \sin^{2} (5 x)) + 32 (\sec^{2} (2 x) (\sec^{2} (2 x) (2 \frac{d}{d x} [x])) + \tan (2 x) \frac{d}{d x} [\sec^{2} (2 x)]) + \frac{d}{d x} [75 \sin^{3} (5 x)]$

Step 3.3.5

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$- 150 (5 \cos^{3} (5 x) - 10 \cos (5 x) \sin^{2} (5 x)) + 32 (\sec^{2} (2 x) (\sec^{2} (2 x) (2 \cdot 1)) + \tan (2 x) \frac{d}{d x} [\sec^{2} (2 x)]) + \frac{d}{d x} [75 \sin^{3} (5 x)]$

Step 3.3.6

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{2}$ and $g (x) = \sec (2 x)$ .

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Step 3.3.6.1

To apply the Chain Rule, set $u_{5}$ as $\sec (2 x)$ .

$- 150 (5 \cos^{3} (5 x) - 10 \cos (5 x) \sin^{2} (5 x)) + 32 (\sec^{2} (2 x) (\sec^{2} (2 x) (2 \cdot 1)) + \tan (2 x) (\frac{d}{d u_{5}} [{u_{5}}^{2}] \frac{d}{d x} [\sec (2 x)])) + \frac{d}{d x} [75 \sin^{3} (5 x)]$

Step 3.3.6.2

Differentiate using the Power Rule which states that $\frac{d}{d u_{5}} [{u_{5}}^{n}]$ is $n {u_{5}}^{n - 1}$ where $n = 2$ .

$- 150 (5 \cos^{3} (5 x) - 10 \cos (5 x) \sin^{2} (5 x)) + 32 (\sec^{2} (2 x) (\sec^{2} (2 x) (2 \cdot 1)) + \tan (2 x) (2 u_{5} \frac{d}{d x} [\sec (2 x)])) + \frac{d}{d x} [75 \sin^{3} (5 x)]$

Step 3.3.6.3

Replace all occurrences of $u_{5}$ with $\sec (2 x)$ .

$- 150 (5 \cos^{3} (5 x) - 10 \cos (5 x) \sin^{2} (5 x)) + 32 (\sec^{2} (2 x) (\sec^{2} (2 x) (2 \cdot 1)) + \tan (2 x) (2 \sec (2 x) \frac{d}{d x} [\sec (2 x)])) + \frac{d}{d x} [75 \sin^{3} (5 x)]$

Step 3.3.7

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \sec (x)$ and $g (x) = 2 x$ .

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Step 3.3.7.1

To apply the Chain Rule, set $u_{6}$ as $2 x$ .

$- 150 (5 \cos^{3} (5 x) - 10 \cos (5 x) \sin^{2} (5 x)) + 32 (\sec^{2} (2 x) (\sec^{2} (2 x) (2 \cdot 1)) + \tan (2 x) (2 \sec (2 x) (\frac{d}{d u_{6}} [\sec (u_{6})] \frac{d}{d x} [2 x]))) + \frac{d}{d x} [75 \sin^{3} (5 x)]$

Step 3.3.7.2

The derivative of $\sec (u_{6})$ with respect to $u_{6}$ is $\sec (u_{6}) \tan (u_{6})$ .

$- 150 (5 \cos^{3} (5 x) - 10 \cos (5 x) \sin^{2} (5 x)) + 32 (\sec^{2} (2 x) (\sec^{2} (2 x) (2 \cdot 1)) + \tan (2 x) (2 \sec (2 x) (\sec (u_{6}) \tan (u_{6}) \frac{d}{d x} [2 x]))) + \frac{d}{d x} [75 \sin^{3} (5 x)]$

Step 3.3.7.3

Replace all occurrences of $u_{6}$ with $2 x$ .

$- 150 (5 \cos^{3} (5 x) - 10 \cos (5 x) \sin^{2} (5 x)) + 32 (\sec^{2} (2 x) (\sec^{2} (2 x) (2 \cdot 1)) + \tan (2 x) (2 \sec (2 x) (\sec (2 x) \tan (2 x) \frac{d}{d x} [2 x]))) + \frac{d}{d x} [75 \sin^{3} (5 x)]$

Step 3.3.8

Since $2$ is constant with respect to $x$ , the derivative of $2 x$ with respect to $x$ is $2 \frac{d}{d x} [x]$ .

$- 150 (5 \cos^{3} (5 x) - 10 \cos (5 x) \sin^{2} (5 x)) + 32 (\sec^{2} (2 x) (\sec^{2} (2 x) (2 \cdot 1)) + \tan (2 x) (2 \sec (2 x) (\sec (2 x) \tan (2 x) (2 \frac{d}{d x} [x])))) + \frac{d}{d x} [75 \sin^{3} (5 x)]$

Step 3.3.9

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$- 150 (5 \cos^{3} (5 x) - 10 \cos (5 x) \sin^{2} (5 x)) + 32 (\sec^{2} (2 x) (\sec^{2} (2 x) (2 \cdot 1)) + \tan (2 x) (2 \sec (2 x) (\sec (2 x) \tan (2 x) (2 \cdot 1)))) + \frac{d}{d x} [75 \sin^{3} (5 x)]$

Step 3.3.10

Multiply $2$ by $1$ .

$- 150 (5 \cos^{3} (5 x) - 10 \cos (5 x) \sin^{2} (5 x)) + 32 (\sec^{2} (2 x) (\sec^{2} (2 x) \cdot 2) + \tan (2 x) (2 \sec (2 x) (\sec (2 x) \tan (2 x) (2 \cdot 1)))) + \frac{d}{d x} [75 \sin^{3} (5 x)]$

Step 3.3.11

Move $2$ to the left of $\sec^{2} (2 x)$ .

$- 150 (5 \cos^{3} (5 x) - 10 \cos (5 x) \sin^{2} (5 x)) + 32 (\sec^{2} (2 x) (2 \cdot \sec^{2} (2 x)) + \tan (2 x) (2 \sec (2 x) (\sec (2 x) \tan (2 x) (2 \cdot 1)))) + \frac{d}{d x} [75 \sin^{3} (5 x)]$

Step 3.3.12

Multiply $\sec^{2} (2 x)$ by $\sec^{2} (2 x)$ by adding the exponents.

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Step 3.3.12.1

Move $\sec^{2} (2 x)$ .

$- 150 (5 \cos^{3} (5 x) - 10 \cos (5 x) \sin^{2} (5 x)) + 32 (\sec^{2} (2 x) \sec^{2} (2 x) \cdot 2 + \tan (2 x) (2 \sec (2 x) (\sec (2 x) \tan (2 x) (2 \cdot 1)))) + \frac{d}{d x} [75 \sin^{3} (5 x)]$

Step 3.3.12.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$- 150 (5 \cos^{3} (5 x) - 10 \cos (5 x) \sin^{2} (5 x)) + 32 ({\sec (2 x)}^{2 + 2} \cdot 2 + \tan (2 x) (2 \sec (2 x) (\sec (2 x) \tan (2 x) (2 \cdot 1)))) + \frac{d}{d x} [75 \sin^{3} (5 x)]$

Step 3.3.12.3

Add $2$ and $2$ .

$- 150 (5 \cos^{3} (5 x) - 10 \cos (5 x) \sin^{2} (5 x)) + 32 (\sec^{4} (2 x) \cdot 2 + \tan (2 x) (2 \sec (2 x) (\sec (2 x) \tan (2 x) (2 \cdot 1)))) + \frac{d}{d x} [75 \sin^{3} (5 x)]$

Step 3.3.13

Move $2$ to the left of $\sec^{4} (2 x)$ .

$- 150 (5 \cos^{3} (5 x) - 10 \cos (5 x) \sin^{2} (5 x)) + 32 (2 \cdot \sec^{4} (2 x) + \tan (2 x) (2 \sec (2 x) (\sec (2 x) \tan (2 x) (2 \cdot 1)))) + \frac{d}{d x} [75 \sin^{3} (5 x)]$

Step 3.3.14

Multiply $2$ by $1$ .

$- 150 (5 \cos^{3} (5 x) - 10 \cos (5 x) \sin^{2} (5 x)) + 32 (2 \sec^{4} (2 x) + \tan (2 x) (2 \sec (2 x) (\sec (2 x) \tan (2 x) \cdot 2))) + \frac{d}{d x} [75 \sin^{3} (5 x)]$

Step 3.3.15

Move $2$ to the left of $\sec (2 x) \tan (2 x)$ .

$- 150 (5 \cos^{3} (5 x) - 10 \cos (5 x) \sin^{2} (5 x)) + 32 (2 \sec^{4} (2 x) + \tan (2 x) (2 \sec (2 x) (2 \cdot (\sec (2 x) \tan (2 x))))) + \frac{d}{d x} [75 \sin^{3} (5 x)]$

Step 3.3.16

Multiply $2$ by $2$ .

$- 150 (5 \cos^{3} (5 x) - 10 \cos (5 x) \sin^{2} (5 x)) + 32 (2 \sec^{4} (2 x) + \tan (2 x) (4 \sec (2 x) (\sec (2 x) \tan (2 x)))) + \frac{d}{d x} [75 \sin^{3} (5 x)]$

Step 3.3.17

Raise $\sec (2 x)$ to the power of $1$ .

$- 150 (5 \cos^{3} (5 x) - 10 \cos (5 x) \sin^{2} (5 x)) + 32 (2 \sec^{4} (2 x) + \tan (2 x) (4 (\sec^{1} (2 x) \sec (2 x)) \tan (2 x))) + \frac{d}{d x} [75 \sin^{3} (5 x)]$

Step 3.3.18

Raise $\sec (2 x)$ to the power of $1$ .

$- 150 (5 \cos^{3} (5 x) - 10 \cos (5 x) \sin^{2} (5 x)) + 32 (2 \sec^{4} (2 x) + \tan (2 x) (4 (\sec^{1} (2 x) \sec^{1} (2 x)) \tan (2 x))) + \frac{d}{d x} [75 \sin^{3} (5 x)]$

Step 3.3.19

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$- 150 (5 \cos^{3} (5 x) - 10 \cos (5 x) \sin^{2} (5 x)) + 32 (2 \sec^{4} (2 x) + \tan (2 x) (4 {\sec (2 x)}^{1 + 1} \tan (2 x))) + \frac{d}{d x} [75 \sin^{3} (5 x)]$

Step 3.3.20

Add $1$ and $1$ .

$- 150 (5 \cos^{3} (5 x) - 10 \cos (5 x) \sin^{2} (5 x)) + 32 (2 \sec^{4} (2 x) + \tan (2 x) (4 \sec^{2} (2 x) \tan (2 x))) + \frac{d}{d x} [75 \sin^{3} (5 x)]$

Step 3.3.21

Raise $\tan (2 x)$ to the power of $1$ .

$- 150 (5 \cos^{3} (5 x) - 10 \cos (5 x) \sin^{2} (5 x)) + 32 (2 \sec^{4} (2 x) + 4 \sec^{2} (2 x) (\tan^{1} (2 x) \tan (2 x))) + \frac{d}{d x} [75 \sin^{3} (5 x)]$

Step 3.3.22

Raise $\tan (2 x)$ to the power of $1$ .

$- 150 (5 \cos^{3} (5 x) - 10 \cos (5 x) \sin^{2} (5 x)) + 32 (2 \sec^{4} (2 x) + 4 \sec^{2} (2 x) (\tan^{1} (2 x) \tan^{1} (2 x))) + \frac{d}{d x} [75 \sin^{3} (5 x)]$

Step 3.3.23

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$- 150 (5 \cos^{3} (5 x) - 10 \cos (5 x) \sin^{2} (5 x)) + 32 (2 \sec^{4} (2 x) + 4 \sec^{2} (2 x) {\tan (2 x)}^{1 + 1}) + \frac{d}{d x} [75 \sin^{3} (5 x)]$

Step 3.3.24

Add $1$ and $1$ .

$- 150 (5 \cos^{3} (5 x) - 10 \cos (5 x) \sin^{2} (5 x)) + 32 (2 \sec^{4} (2 x) + 4 \sec^{2} (2 x) \tan^{2} (2 x)) + \frac{d}{d x} [75 \sin^{3} (5 x)]$

Step 3.4

Evaluate $\frac{d}{d x} [75 \sin^{3} (5 x)]$ .

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Step 3.4.1

Since $75$ is constant with respect to $x$ , the derivative of $75 \sin^{3} (5 x)$ with respect to $x$ is $75 \frac{d}{d x} [\sin^{3} (5 x)]$ .

$- 150 (5 \cos^{3} (5 x) - 10 \cos (5 x) \sin^{2} (5 x)) + 32 (2 \sec^{4} (2 x) + 4 \sec^{2} (2 x) \tan^{2} (2 x)) + 75 \frac{d}{d x} [\sin^{3} (5 x)]$

Step 3.4.2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{3}$ and $g (x) = \sin (5 x)$ .

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Step 3.4.2.1

To apply the Chain Rule, set $u_{7}$ as $\sin (5 x)$ .

$- 150 (5 \cos^{3} (5 x) - 10 \cos (5 x) \sin^{2} (5 x)) + 32 (2 \sec^{4} (2 x) + 4 \sec^{2} (2 x) \tan^{2} (2 x)) + 75 (\frac{d}{d u_{7}} [{u_{7}}^{3}] \frac{d}{d x} [\sin (5 x)])$

Step 3.4.2.2

Differentiate using the Power Rule which states that $\frac{d}{d u_{7}} [{u_{7}}^{n}]$ is $n {u_{7}}^{n - 1}$ where $n = 3$ .

$- 150 (5 \cos^{3} (5 x) - 10 \cos (5 x) \sin^{2} (5 x)) + 32 (2 \sec^{4} (2 x) + 4 \sec^{2} (2 x) \tan^{2} (2 x)) + 75 (3 {u_{7}}^{2} \frac{d}{d x} [\sin (5 x)])$

Step 3.4.2.3

Replace all occurrences of $u_{7}$ with $\sin (5 x)$ .

$- 150 (5 \cos^{3} (5 x) - 10 \cos (5 x) \sin^{2} (5 x)) + 32 (2 \sec^{4} (2 x) + 4 \sec^{2} (2 x) \tan^{2} (2 x)) + 75 (3 \sin^{2} (5 x) \frac{d}{d x} [\sin (5 x)])$

Step 3.4.3

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \sin (x)$ and $g (x) = 5 x$ .

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Step 3.4.3.1

To apply the Chain Rule, set $u_{8}$ as $5 x$ .

$- 150 (5 \cos^{3} (5 x) - 10 \cos (5 x) \sin^{2} (5 x)) + 32 (2 \sec^{4} (2 x) + 4 \sec^{2} (2 x) \tan^{2} (2 x)) + 75 (3 \sin^{2} (5 x) (\frac{d}{d u_{8}} [\sin (u_{8})] \frac{d}{d x} [5 x]))$

Step 3.4.3.2

The derivative of $\sin (u_{8})$ with respect to $u_{8}$ is $\cos (u_{8})$ .

$- 150 (5 \cos^{3} (5 x) - 10 \cos (5 x) \sin^{2} (5 x)) + 32 (2 \sec^{4} (2 x) + 4 \sec^{2} (2 x) \tan^{2} (2 x)) + 75 (3 \sin^{2} (5 x) (\cos (u_{8}) \frac{d}{d x} [5 x]))$

Step 3.4.3.3

Replace all occurrences of $u_{8}$ with $5 x$ .

$- 150 (5 \cos^{3} (5 x) - 10 \cos (5 x) \sin^{2} (5 x)) + 32 (2 \sec^{4} (2 x) + 4 \sec^{2} (2 x) \tan^{2} (2 x)) + 75 (3 \sin^{2} (5 x) (\cos (5 x) \frac{d}{d x} [5 x]))$

Step 3.4.4

Since $5$ is constant with respect to $x$ , the derivative of $5 x$ with respect to $x$ is $5 \frac{d}{d x} [x]$ .

$- 150 (5 \cos^{3} (5 x) - 10 \cos (5 x) \sin^{2} (5 x)) + 32 (2 \sec^{4} (2 x) + 4 \sec^{2} (2 x) \tan^{2} (2 x)) + 75 (3 \sin^{2} (5 x) (\cos (5 x) (5 \frac{d}{d x} [x])))$

Step 3.4.5

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$- 150 (5 \cos^{3} (5 x) - 10 \cos (5 x) \sin^{2} (5 x)) + 32 (2 \sec^{4} (2 x) + 4 \sec^{2} (2 x) \tan^{2} (2 x)) + 75 (3 \sin^{2} (5 x) (\cos (5 x) (5 \cdot 1)))$

Step 3.4.6

Multiply $5$ by $1$ .

$- 150 (5 \cos^{3} (5 x) - 10 \cos (5 x) \sin^{2} (5 x)) + 32 (2 \sec^{4} (2 x) + 4 \sec^{2} (2 x) \tan^{2} (2 x)) + 75 (3 \sin^{2} (5 x) (\cos (5 x) \cdot 5))$

Step 3.4.7

Move $5$ to the left of $\cos (5 x)$ .

$- 150 (5 \cos^{3} (5 x) - 10 \cos (5 x) \sin^{2} (5 x)) + 32 (2 \sec^{4} (2 x) + 4 \sec^{2} (2 x) \tan^{2} (2 x)) + 75 (3 \sin^{2} (5 x) (5 \cdot \cos (5 x)))$

Step 3.4.8

Multiply $5$ by $3$ .

$- 150 (5 \cos^{3} (5 x) - 10 \cos (5 x) \sin^{2} (5 x)) + 32 (2 \sec^{4} (2 x) + 4 \sec^{2} (2 x) \tan^{2} (2 x)) + 75 (15 \sin^{2} (5 x) \cos (5 x))$

Step 3.4.9

Multiply $15$ by $75$ .

$- 150 (5 \cos^{3} (5 x) - 10 \cos (5 x) \sin^{2} (5 x)) + 32 (2 \sec^{4} (2 x) + 4 \sec^{2} (2 x) \tan^{2} (2 x)) + 1125 \sin^{2} (5 x) \cos (5 x)$

Step 3.5

Simplify.

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Step 3.5.1

Apply the distributive property.

$- 150 (5 \cos^{3} (5 x)) - 150 (- 10 \cos (5 x) \sin^{2} (5 x)) + 32 (2 \sec^{4} (2 x) + 4 \sec^{2} (2 x) \tan^{2} (2 x)) + 1125 \sin^{2} (5 x) \cos (5 x)$

Step 3.5.2

Apply the distributive property.

$- 150 (5 \cos^{3} (5 x)) - 150 (- 10 \cos (5 x) \sin^{2} (5 x)) + 32 (2 \sec^{4} (2 x)) + 32 (4 \sec^{2} (2 x) \tan^{2} (2 x)) + 1125 \sin^{2} (5 x) \cos (5 x)$

Step 3.5.3

Combine terms.

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Step 3.5.3.1

Multiply $5$ by $- 150$ .

$- 750 \cos^{3} (5 x) - 150 (- 10 \cos (5 x) \sin^{2} (5 x)) + 32 (2 \sec^{4} (2 x)) + 32 (4 \sec^{2} (2 x) \tan^{2} (2 x)) + 1125 \sin^{2} (5 x) \cos (5 x)$

Step 3.5.3.2

Multiply $- 10$ by $- 150$ .

$- 750 \cos^{3} (5 x) + 1500 (\cos (5 x) \sin^{2} (5 x)) + 32 (2 \sec^{4} (2 x)) + 32 (4 \sec^{2} (2 x) \tan^{2} (2 x)) + 1125 \sin^{2} (5 x) \cos (5 x)$

Step 3.5.3.3

Multiply $2$ by $32$ .

$- 750 \cos^{3} (5 x) + 1500 \cos (5 x) \sin^{2} (5 x) + 64 \sec^{4} (2 x) + 32 (4 \sec^{2} (2 x) \tan^{2} (2 x)) + 1125 \sin^{2} (5 x) \cos (5 x)$

Step 3.5.3.4

Multiply $4$ by $32$ .

$- 750 \cos^{3} (5 x) + 1500 \cos (5 x) \sin^{2} (5 x) + 64 \sec^{4} (2 x) + 128 (\sec^{2} (2 x) \tan^{2} (2 x)) + 1125 \sin^{2} (5 x) \cos (5 x)$

Step 3.5.3.5

Reorder the factors of $1500 \cos (5 x) \sin^{2} (5 x)$ .

$- 750 \cos^{3} (5 x) + 1500 \sin^{2} (5 x) \cos (5 x) + 64 \sec^{4} (2 x) + 128 \sec^{2} (2 x) \tan^{2} (2 x) + 1125 \sin^{2} (5 x) \cos (5 x)$

Step 3.5.3.6

Add $1500 \sin^{2} (5 x) \cos (5 x)$ and $1125 \sin^{2} (5 x) \cos (5 x)$ .

$- 750 \cos^{3} (5 x) + 2625 \sin^{2} (5 x) \cos (5 x) + 64 \sec^{4} (2 x) + 128 \sec^{2} (2 x) \tan^{2} (2 x)$

Step 3.5.4

Reorder terms.

$2625 \sin^{2} (5 x) \cos (5 x) + 128 \sec^{2} (2 x) \tan^{2} (2 x) - 750 \cos^{3} (5 x) + 64 \sec^{4} (2 x)$

Step 3.5.5

Simplify each term.

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Step 3.5.5.1

Rewrite $\sec (2 x)$ in terms of sines and cosines.

$2625 \sin^{2} (5 x) \cos (5 x) + 128 {(\frac{1}{\cos (2 x)})}^{2} \tan^{2} (2 x) - 750 \cos^{3} (5 x) + 64 \sec^{4} (2 x)$

Step 3.5.5.2

Apply the product rule to $\frac{1}{\cos (2 x)}$ .

$2625 \sin^{2} (5 x) \cos (5 x) + 128 \frac{1^{2}}{\cos^{2} (2 x)} \tan^{2} (2 x) - 750 \cos^{3} (5 x) + 64 \sec^{4} (2 x)$

Step 3.5.5.3

One to any power is one.

$2625 \sin^{2} (5 x) \cos (5 x) + 128 \frac{1}{\cos^{2} (2 x)} \tan^{2} (2 x) - 750 \cos^{3} (5 x) + 64 \sec^{4} (2 x)$

Step 3.5.5.4

Combine $128$ and $\frac{1}{\cos^{2} (2 x)}$ .

$2625 \sin^{2} (5 x) \cos (5 x) + \frac{128}{\cos^{2} (2 x)} \tan^{2} (2 x) - 750 \cos^{3} (5 x) + 64 \sec^{4} (2 x)$

Step 3.5.5.5

Rewrite $\tan (2 x)$ in terms of sines and cosines.

$2625 \sin^{2} (5 x) \cos (5 x) + \frac{128}{\cos^{2} (2 x)} {(\frac{\sin (2 x)}{\cos (2 x)})}^{2} - 750 \cos^{3} (5 x) + 64 \sec^{4} (2 x)$

Step 3.5.5.6

Apply the product rule to $\frac{\sin (2 x)}{\cos (2 x)}$ .

$2625 \sin^{2} (5 x) \cos (5 x) + \frac{128}{\cos^{2} (2 x)} \cdot \frac{\sin^{2} (2 x)}{\cos^{2} (2 x)} - 750 \cos^{3} (5 x) + 64 \sec^{4} (2 x)$

Step 3.5.5.7

Combine.

$2625 \sin^{2} (5 x) \cos (5 x) + \frac{128 \sin^{2} (2 x)}{\cos^{2} (2 x) \cos^{2} (2 x)} - 750 \cos^{3} (5 x) + 64 \sec^{4} (2 x)$

Step 3.5.5.8

Multiply $\cos^{2} (2 x)$ by $\cos^{2} (2 x)$ by adding the exponents.

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Step 3.5.5.8.1

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$2625 \sin^{2} (5 x) \cos (5 x) + \frac{128 \sin^{2} (2 x)}{{\cos (2 x)}^{2 + 2}} - 750 \cos^{3} (5 x) + 64 \sec^{4} (2 x)$

Step 3.5.5.8.2

Add $2$ and $2$ .

$2625 \sin^{2} (5 x) \cos (5 x) + \frac{128 \sin^{2} (2 x)}{\cos^{4} (2 x)} - 750 \cos^{3} (5 x) + 64 \sec^{4} (2 x)$

Step 3.5.5.9

Rewrite $\sec (2 x)$ in terms of sines and cosines.

$2625 \sin^{2} (5 x) \cos (5 x) + \frac{128 \sin^{2} (2 x)}{\cos^{4} (2 x)} - 750 \cos^{3} (5 x) + 64 {(\frac{1}{\cos (2 x)})}^{4}$

Step 3.5.5.10

Apply the product rule to $\frac{1}{\cos (2 x)}$ .

$2625 \sin^{2} (5 x) \cos (5 x) + \frac{128 \sin^{2} (2 x)}{\cos^{4} (2 x)} - 750 \cos^{3} (5 x) + 64 \frac{1^{4}}{\cos^{4} (2 x)}$

Step 3.5.5.11

One to any power is one.

$2625 \sin^{2} (5 x) \cos (5 x) + \frac{128 \sin^{2} (2 x)}{\cos^{4} (2 x)} - 750 \cos^{3} (5 x) + 64 \frac{1}{\cos^{4} (2 x)}$

Step 3.5.5.12

Combine $64$ and $\frac{1}{\cos^{4} (2 x)}$ .

$2625 \sin^{2} (5 x) \cos (5 x) + \frac{128 \sin^{2} (2 x)}{\cos^{4} (2 x)} - 750 \cos^{3} (5 x) + \frac{64}{\cos^{4} (2 x)}$

Step 3.5.6

Simplify each term.

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Step 3.5.6.1

Multiply by $1$ .

$2625 \sin^{2} (5 x) \cos (5 x) + \frac{128 (\sin^{2} (2 x) \cdot 1)}{\cos^{4} (2 x)} - 750 \cos^{3} (5 x) + \frac{64}{\cos^{4} (2 x)}$

Step 3.5.6.2

Factor $\cos^{2} (2 x)$ out of $\cos^{4} (2 x)$ .

$2625 \sin^{2} (5 x) \cos (5 x) + \frac{128 (\sin^{2} (2 x) \cdot 1)}{\cos^{2} (2 x) \cos^{2} (2 x)} - 750 \cos^{3} (5 x) + \frac{64}{\cos^{4} (2 x)}$

Step 3.5.6.3

Separate fractions.

$2625 \sin^{2} (5 x) \cos (5 x) + \frac{128 \cdot (1)}{\cos^{2} (2 x)} \cdot \frac{\sin^{2} (2 x)}{\cos^{2} (2 x)} - 750 \cos^{3} (5 x) + \frac{64}{\cos^{4} (2 x)}$

Step 3.5.6.4

Convert from $\frac{\sin^{2} (2 x)}{\cos^{2} (2 x)}$ to $\tan^{2} (2 x)$ .

$2625 \sin^{2} (5 x) \cos (5 x) + \frac{128 \cdot (1)}{\cos^{2} (2 x)} \tan^{2} (2 x) - 750 \cos^{3} (5 x) + \frac{64}{\cos^{4} (2 x)}$

Step 3.5.6.5

Multiply $128$ by $1$ .

$2625 \sin^{2} (5 x) \cos (5 x) + \frac{128}{\cos^{2} (2 x)} \tan^{2} (2 x) - 750 \cos^{3} (5 x) + \frac{64}{\cos^{4} (2 x)}$

Step 3.5.6.6

Multiply by $1$ .

$2625 \sin^{2} (5 x) \cos (5 x) + \frac{128}{\cos^{2} (2 x) \cdot 1} \tan^{2} (2 x) - 750 \cos^{3} (5 x) + \frac{64}{\cos^{4} (2 x)}$

Step 3.5.6.7

Separate fractions.

$2625 \sin^{2} (5 x) \cos (5 x) + \frac{128}{1} \cdot \frac{1}{\cos^{2} (2 x)} \tan^{2} (2 x) - 750 \cos^{3} (5 x) + \frac{64}{\cos^{4} (2 x)}$

Step 3.5.6.8

Convert from $\frac{1}{\cos^{2} (2 x)}$ to $\sec^{2} (2 x)$ .

$2625 \sin^{2} (5 x) \cos (5 x) + \frac{128}{1} \sec^{2} (2 x) \tan^{2} (2 x) - 750 \cos^{3} (5 x) + \frac{64}{\cos^{4} (2 x)}$

Step 3.5.6.9

Divide $128$ by $1$ .

$2625 \sin^{2} (5 x) \cos (5 x) + 128 \sec^{2} (2 x) \tan^{2} (2 x) - 750 \cos^{3} (5 x) + \frac{64}{\cos^{4} (2 x)}$

Step 3.5.6.10

Multiply by $1$ .

$2625 \sin^{2} (5 x) \cos (5 x) + 128 \sec^{2} (2 x) \tan^{2} (2 x) - 750 \cos^{3} (5 x) + \frac{64}{\cos^{4} (2 x) \cdot 1}$

Step 3.5.6.11

Separate fractions.

$2625 \sin^{2} (5 x) \cos (5 x) + 128 \sec^{2} (2 x) \tan^{2} (2 x) - 750 \cos^{3} (5 x) + \frac{64}{1} \cdot \frac{1}{\cos^{4} (2 x)}$

Step 3.5.6.12

Convert from $\frac{1}{\cos^{4} (2 x)}$ to $\sec^{4} (2 x)$ .

$2625 \sin^{2} (5 x) \cos (5 x) + 128 \sec^{2} (2 x) \tan^{2} (2 x) - 750 \cos^{3} (5 x) + \frac{64}{1} \sec^{4} (2 x)$

Step 3.5.6.13

Divide $64$ by $1$ .

$f''' (x) = 2625 \sin^{2} (5 x) \cos (5 x) + 128 \sec^{2} (2 x) \tan^{2} (2 x) - 750 \cos^{3} (5 x) + 64 \sec^{4} (2 x)$

Step 4

Find the fourth derivative.

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Step 4.1

By the Sum Rule, the derivative of $2625 \sin^{2} (5 x) \cos (5 x) + 128 \sec^{2} (2 x) \tan^{2} (2 x) - 750 \cos^{3} (5 x) + 64 \sec^{4} (2 x)$ with respect to $x$ is $\frac{d}{d x} [2625 \sin^{2} (5 x) \cos (5 x)] + \frac{d}{d x} [128 \sec^{2} (2 x) \tan^{2} (2 x)] + \frac{d}{d x} [- 750 \cos^{3} (5 x)] + \frac{d}{d x} [64 \sec^{4} (2 x)]$ .

$\frac{d}{d x} [2625 \sin^{2} (5 x) \cos (5 x)] + \frac{d}{d x} [128 \sec^{2} (2 x) \tan^{2} (2 x)] + \frac{d}{d x} [- 750 \cos^{3} (5 x)] + \frac{d}{d x} [64 \sec^{4} (2 x)]$

Step 4.2

Evaluate $\frac{d}{d x} [2625 \sin^{2} (5 x) \cos (5 x)]$ .

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Step 4.2.1

Since $2625$ is constant with respect to $x$ , the derivative of $2625 \sin^{2} (5 x) \cos (5 x)$ with respect to $x$ is $2625 \frac{d}{d x} [\sin^{2} (5 x) \cos (5 x)]$ .

$2625 \frac{d}{d x} [\sin^{2} (5 x) \cos (5 x)] + \frac{d}{d x} [128 \sec^{2} (2 x) \tan^{2} (2 x)] + \frac{d}{d x} [- 750 \cos^{3} (5 x)] + \frac{d}{d x} [64 \sec^{4} (2 x)]$

Step 4.2.2

$2625 (\sin^{2} (5 x) \frac{d}{d x} [\cos (5 x)] + \cos (5 x) \frac{d}{d x} [\sin^{2} (5 x)]) + \frac{d}{d x} [128 \sec^{2} (2 x) \tan^{2} (2 x)] + \frac{d}{d x} [- 750 \cos^{3} (5 x)] + \frac{d}{d x} [64 \sec^{4} (2 x)]$

Step 4.2.3

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \cos (x)$ and $g (x) = 5 x$ .

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Step 4.2.3.1

To apply the Chain Rule, set $u_{1}$ as $5 x$ .

$2625 (\sin^{2} (5 x) (\frac{d}{d u_{1}} [\cos (u_{1})] \frac{d}{d x} [5 x]) + \cos (5 x) \frac{d}{d x} [\sin^{2} (5 x)]) + \frac{d}{d x} [128 \sec^{2} (2 x) \tan^{2} (2 x)] + \frac{d}{d x} [- 750 \cos^{3} (5 x)] + \frac{d}{d x} [64 \sec^{4} (2 x)]$

Step 4.2.3.2

The derivative of $\cos (u_{1})$ with respect to $u_{1}$ is $- \sin (u_{1})$ .

$2625 (\sin^{2} (5 x) (- \sin (u_{1}) \frac{d}{d x} [5 x]) + \cos (5 x) \frac{d}{d x} [\sin^{2} (5 x)]) + \frac{d}{d x} [128 \sec^{2} (2 x) \tan^{2} (2 x)] + \frac{d}{d x} [- 750 \cos^{3} (5 x)] + \frac{d}{d x} [64 \sec^{4} (2 x)]$

Step 4.2.3.3

Replace all occurrences of $u_{1}$ with $5 x$ .

$2625 (\sin^{2} (5 x) (- \sin (5 x) \frac{d}{d x} [5 x]) + \cos (5 x) \frac{d}{d x} [\sin^{2} (5 x)]) + \frac{d}{d x} [128 \sec^{2} (2 x) \tan^{2} (2 x)] + \frac{d}{d x} [- 750 \cos^{3} (5 x)] + \frac{d}{d x} [64 \sec^{4} (2 x)]$

Step 4.2.4

Since $5$ is constant with respect to $x$ , the derivative of $5 x$ with respect to $x$ is $5 \frac{d}{d x} [x]$ .

$2625 (\sin^{2} (5 x) (- \sin (5 x) (5 \frac{d}{d x} [x])) + \cos (5 x) \frac{d}{d x} [\sin^{2} (5 x)]) + \frac{d}{d x} [128 \sec^{2} (2 x) \tan^{2} (2 x)] + \frac{d}{d x} [- 750 \cos^{3} (5 x)] + \frac{d}{d x} [64 \sec^{4} (2 x)]$

Step 4.2.5

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$2625 (\sin^{2} (5 x) (- \sin (5 x) (5 \cdot 1)) + \cos (5 x) \frac{d}{d x} [\sin^{2} (5 x)]) + \frac{d}{d x} [128 \sec^{2} (2 x) \tan^{2} (2 x)] + \frac{d}{d x} [- 750 \cos^{3} (5 x)] + \frac{d}{d x} [64 \sec^{4} (2 x)]$

Step 4.2.6

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{2}$ and $g (x) = \sin (5 x)$ .

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Step 4.2.6.1

To apply the Chain Rule, set $u_{2}$ as $\sin (5 x)$ .

$2625 (\sin^{2} (5 x) (- \sin (5 x) (5 \cdot 1)) + \cos (5 x) (\frac{d}{d u_{2}} [{u_{2}}^{2}] \frac{d}{d x} [\sin (5 x)])) + \frac{d}{d x} [128 \sec^{2} (2 x) \tan^{2} (2 x)] + \frac{d}{d x} [- 750 \cos^{3} (5 x)] + \frac{d}{d x} [64 \sec^{4} (2 x)]$

Step 4.2.6.2

Differentiate using the Power Rule which states that $\frac{d}{d u_{2}} [{u_{2}}^{n}]$ is $n {u_{2}}^{n - 1}$ where $n = 2$ .

$2625 (\sin^{2} (5 x) (- \sin (5 x) (5 \cdot 1)) + \cos (5 x) (2 u_{2} \frac{d}{d x} [\sin (5 x)])) + \frac{d}{d x} [128 \sec^{2} (2 x) \tan^{2} (2 x)] + \frac{d}{d x} [- 750 \cos^{3} (5 x)] + \frac{d}{d x} [64 \sec^{4} (2 x)]$

Step 4.2.6.3

Replace all occurrences of $u_{2}$ with $\sin (5 x)$ .

$2625 (\sin^{2} (5 x) (- \sin (5 x) (5 \cdot 1)) + \cos (5 x) (2 \sin (5 x) \frac{d}{d x} [\sin (5 x)])) + \frac{d}{d x} [128 \sec^{2} (2 x) \tan^{2} (2 x)] + \frac{d}{d x} [- 750 \cos^{3} (5 x)] + \frac{d}{d x} [64 \sec^{4} (2 x)]$

Step 4.2.7

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \sin (x)$ and $g (x) = 5 x$ .

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Step 4.2.7.1

To apply the Chain Rule, set $u_{3}$ as $5 x$ .

$2625 (\sin^{2} (5 x) (- \sin (5 x) (5 \cdot 1)) + \cos (5 x) (2 \sin (5 x) (\frac{d}{d u_{3}} [\sin (u_{3})] \frac{d}{d x} [5 x]))) + \frac{d}{d x} [128 \sec^{2} (2 x) \tan^{2} (2 x)] + \frac{d}{d x} [- 750 \cos^{3} (5 x)] + \frac{d}{d x} [64 \sec^{4} (2 x)]$

Step 4.2.7.2

The derivative of $\sin (u_{3})$ with respect to $u_{3}$ is $\cos (u_{3})$ .

$2625 (\sin^{2} (5 x) (- \sin (5 x) (5 \cdot 1)) + \cos (5 x) (2 \sin (5 x) (\cos (u_{3}) \frac{d}{d x} [5 x]))) + \frac{d}{d x} [128 \sec^{2} (2 x) \tan^{2} (2 x)] + \frac{d}{d x} [- 750 \cos^{3} (5 x)] + \frac{d}{d x} [64 \sec^{4} (2 x)]$

Step 4.2.7.3

Replace all occurrences of $u_{3}$ with $5 x$ .

$2625 (\sin^{2} (5 x) (- \sin (5 x) (5 \cdot 1)) + \cos (5 x) (2 \sin (5 x) (\cos (5 x) \frac{d}{d x} [5 x]))) + \frac{d}{d x} [128 \sec^{2} (2 x) \tan^{2} (2 x)] + \frac{d}{d x} [- 750 \cos^{3} (5 x)] + \frac{d}{d x} [64 \sec^{4} (2 x)]$

Step 4.2.8

Since $5$ is constant with respect to $x$ , the derivative of $5 x$ with respect to $x$ is $5 \frac{d}{d x} [x]$ .

$2625 (\sin^{2} (5 x) (- \sin (5 x) (5 \cdot 1)) + \cos (5 x) (2 \sin (5 x) (\cos (5 x) (5 \frac{d}{d x} [x])))) + \frac{d}{d x} [128 \sec^{2} (2 x) \tan^{2} (2 x)] + \frac{d}{d x} [- 750 \cos^{3} (5 x)] + \frac{d}{d x} [64 \sec^{4} (2 x)]$

Step 4.2.9

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$2625 (\sin^{2} (5 x) (- \sin (5 x) (5 \cdot 1)) + \cos (5 x) (2 \sin (5 x) (\cos (5 x) (5 \cdot 1)))) + \frac{d}{d x} [128 \sec^{2} (2 x) \tan^{2} (2 x)] + \frac{d}{d x} [- 750 \cos^{3} (5 x)] + \frac{d}{d x} [64 \sec^{4} (2 x)]$

Step 4.2.10

Multiply $5$ by $1$ .

$2625 (\sin^{2} (5 x) (- \sin (5 x) \cdot 5) + \cos (5 x) (2 \sin (5 x) (\cos (5 x) (5 \cdot 1)))) + \frac{d}{d x} [128 \sec^{2} (2 x) \tan^{2} (2 x)] + \frac{d}{d x} [- 750 \cos^{3} (5 x)] + \frac{d}{d x} [64 \sec^{4} (2 x)]$

Step 4.2.11

Multiply $5$ by $- 1$ .

$2625 (\sin^{2} (5 x) (- 5 \sin (5 x)) + \cos (5 x) (2 \sin (5 x) (\cos (5 x) (5 \cdot 1)))) + \frac{d}{d x} [128 \sec^{2} (2 x) \tan^{2} (2 x)] + \frac{d}{d x} [- 750 \cos^{3} (5 x)] + \frac{d}{d x} [64 \sec^{4} (2 x)]$

Step 4.2.12

Multiply $\sin^{2} (5 x)$ by $\sin (5 x)$ by adding the exponents.

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Step 4.2.12.1

Move $\sin (5 x)$ .

$2625 (\sin (5 x) \sin^{2} (5 x) \cdot - 5 + \cos (5 x) (2 \sin (5 x) (\cos (5 x) (5 \cdot 1)))) + \frac{d}{d x} [128 \sec^{2} (2 x) \tan^{2} (2 x)] + \frac{d}{d x} [- 750 \cos^{3} (5 x)] + \frac{d}{d x} [64 \sec^{4} (2 x)]$

Step 4.2.12.2

Multiply $\sin (5 x)$ by $\sin^{2} (5 x)$ .

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Step 4.2.12.2.1

Raise $\sin (5 x)$ to the power of $1$ .

$2625 (\sin^{1} (5 x) \sin^{2} (5 x) \cdot - 5 + \cos (5 x) (2 \sin (5 x) (\cos (5 x) (5 \cdot 1)))) + \frac{d}{d x} [128 \sec^{2} (2 x) \tan^{2} (2 x)] + \frac{d}{d x} [- 750 \cos^{3} (5 x)] + \frac{d}{d x} [64 \sec^{4} (2 x)]$

Step 4.2.12.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$2625 ({\sin (5 x)}^{1 + 2} \cdot - 5 + \cos (5 x) (2 \sin (5 x) (\cos (5 x) (5 \cdot 1)))) + \frac{d}{d x} [128 \sec^{2} (2 x) \tan^{2} (2 x)] + \frac{d}{d x} [- 750 \cos^{3} (5 x)] + \frac{d}{d x} [64 \sec^{4} (2 x)]$

Step 4.2.12.3

Add $1$ and $2$ .

$2625 (\sin^{3} (5 x) \cdot - 5 + \cos (5 x) (2 \sin (5 x) (\cos (5 x) (5 \cdot 1)))) + \frac{d}{d x} [128 \sec^{2} (2 x) \tan^{2} (2 x)] + \frac{d}{d x} [- 750 \cos^{3} (5 x)] + \frac{d}{d x} [64 \sec^{4} (2 x)]$

Step 4.2.13

Move $- 5$ to the left of $\sin^{3} (5 x)$ .

$2625 (- 5 \cdot \sin^{3} (5 x) + \cos (5 x) (2 \sin (5 x) (\cos (5 x) (5 \cdot 1)))) + \frac{d}{d x} [128 \sec^{2} (2 x) \tan^{2} (2 x)] + \frac{d}{d x} [- 750 \cos^{3} (5 x)] + \frac{d}{d x} [64 \sec^{4} (2 x)]$

Step 4.2.14

Multiply $5$ by $1$ .

$2625 (- 5 \sin^{3} (5 x) + \cos (5 x) (2 \sin (5 x) (\cos (5 x) \cdot 5))) + \frac{d}{d x} [128 \sec^{2} (2 x) \tan^{2} (2 x)] + \frac{d}{d x} [- 750 \cos^{3} (5 x)] + \frac{d}{d x} [64 \sec^{4} (2 x)]$

Step 4.2.15

Move $5$ to the left of $\cos (5 x)$ .

$2625 (- 5 \sin^{3} (5 x) + \cos (5 x) (2 \sin (5 x) (5 \cdot \cos (5 x)))) + \frac{d}{d x} [128 \sec^{2} (2 x) \tan^{2} (2 x)] + \frac{d}{d x} [- 750 \cos^{3} (5 x)] + \frac{d}{d x} [64 \sec^{4} (2 x)]$

Step 4.2.16

Multiply $5$ by $2$ .

$2625 (- 5 \sin^{3} (5 x) + \cos (5 x) (10 \sin (5 x) \cos (5 x))) + \frac{d}{d x} [128 \sec^{2} (2 x) \tan^{2} (2 x)] + \frac{d}{d x} [- 750 \cos^{3} (5 x)] + \frac{d}{d x} [64 \sec^{4} (2 x)]$

Step 4.2.17

Raise $\cos (5 x)$ to the power of $1$ .

$2625 (- 5 \sin^{3} (5 x) + 10 \sin (5 x) (\cos^{1} (5 x) \cos (5 x))) + \frac{d}{d x} [128 \sec^{2} (2 x) \tan^{2} (2 x)] + \frac{d}{d x} [- 750 \cos^{3} (5 x)] + \frac{d}{d x} [64 \sec^{4} (2 x)]$

Step 4.2.18

Raise $\cos (5 x)$ to the power of $1$ .

$2625 (- 5 \sin^{3} (5 x) + 10 \sin (5 x) (\cos^{1} (5 x) \cos^{1} (5 x))) + \frac{d}{d x} [128 \sec^{2} (2 x) \tan^{2} (2 x)] + \frac{d}{d x} [- 750 \cos^{3} (5 x)] + \frac{d}{d x} [64 \sec^{4} (2 x)]$

Step 4.2.19

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$2625 (- 5 \sin^{3} (5 x) + 10 \sin (5 x) {\cos (5 x)}^{1 + 1}) + \frac{d}{d x} [128 \sec^{2} (2 x) \tan^{2} (2 x)] + \frac{d}{d x} [- 750 \cos^{3} (5 x)] + \frac{d}{d x} [64 \sec^{4} (2 x)]$

Step 4.2.20

Add $1$ and $1$ .

$2625 (- 5 \sin^{3} (5 x) + 10 \sin (5 x) \cos^{2} (5 x)) + \frac{d}{d x} [128 \sec^{2} (2 x) \tan^{2} (2 x)] + \frac{d}{d x} [- 750 \cos^{3} (5 x)] + \frac{d}{d x} [64 \sec^{4} (2 x)]$

Step 4.3

Evaluate $\frac{d}{d x} [128 \sec^{2} (2 x) \tan^{2} (2 x)]$ .

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Step 4.3.1

Since $128$ is constant with respect to $x$ , the derivative of $128 \sec^{2} (2 x) \tan^{2} (2 x)$ with respect to $x$ is $128 \frac{d}{d x} [\sec^{2} (2 x) \tan^{2} (2 x)]$ .

$2625 (- 5 \sin^{3} (5 x) + 10 \sin (5 x) \cos^{2} (5 x)) + 128 \frac{d}{d x} [\sec^{2} (2 x) \tan^{2} (2 x)] + \frac{d}{d x} [- 750 \cos^{3} (5 x)] + \frac{d}{d x} [64 \sec^{4} (2 x)]$

Step 4.3.2

$2625 (- 5 \sin^{3} (5 x) + 10 \sin (5 x) \cos^{2} (5 x)) + 128 (\sec^{2} (2 x) \frac{d}{d x} [\tan^{2} (2 x)] + \tan^{2} (2 x) \frac{d}{d x} [\sec^{2} (2 x)]) + \frac{d}{d x} [- 750 \cos^{3} (5 x)] + \frac{d}{d x} [64 \sec^{4} (2 x)]$

Step 4.3.3

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{2}$ and $g (x) = \tan (2 x)$ .

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Step 4.3.3.1

To apply the Chain Rule, set $u_{4}$ as $\tan (2 x)$ .

$2625 (- 5 \sin^{3} (5 x) + 10 \sin (5 x) \cos^{2} (5 x)) + 128 (\sec^{2} (2 x) (\frac{d}{d u_{4}} [{u_{4}}^{2}] \frac{d}{d x} [\tan (2 x)]) + \tan^{2} (2 x) \frac{d}{d x} [\sec^{2} (2 x)]) + \frac{d}{d x} [- 750 \cos^{3} (5 x)] + \frac{d}{d x} [64 \sec^{4} (2 x)]$

Step 4.3.3.2

Differentiate using the Power Rule which states that $\frac{d}{d u_{4}} [{u_{4}}^{n}]$ is $n {u_{4}}^{n - 1}$ where $n = 2$ .

$2625 (- 5 \sin^{3} (5 x) + 10 \sin (5 x) \cos^{2} (5 x)) + 128 (\sec^{2} (2 x) (2 u_{4} \frac{d}{d x} [\tan (2 x)]) + \tan^{2} (2 x) \frac{d}{d x} [\sec^{2} (2 x)]) + \frac{d}{d x} [- 750 \cos^{3} (5 x)] + \frac{d}{d x} [64 \sec^{4} (2 x)]$

Step 4.3.3.3

Replace all occurrences of $u_{4}$ with $\tan (2 x)$ .

$2625 (- 5 \sin^{3} (5 x) + 10 \sin (5 x) \cos^{2} (5 x)) + 128 (\sec^{2} (2 x) (2 \tan (2 x) \frac{d}{d x} [\tan (2 x)]) + \tan^{2} (2 x) \frac{d}{d x} [\sec^{2} (2 x)]) + \frac{d}{d x} [- 750 \cos^{3} (5 x)] + \frac{d}{d x} [64 \sec^{4} (2 x)]$

Step 4.3.4

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \tan (x)$ and $g (x) = 2 x$ .

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Step 4.3.4.1

To apply the Chain Rule, set $u_{5}$ as $2 x$ .

$2625 (- 5 \sin^{3} (5 x) + 10 \sin (5 x) \cos^{2} (5 x)) + 128 (\sec^{2} (2 x) (2 \tan (2 x) (\frac{d}{d u_{5}} [\tan (u_{5})] \frac{d}{d x} [2 x])) + \tan^{2} (2 x) \frac{d}{d x} [\sec^{2} (2 x)]) + \frac{d}{d x} [- 750 \cos^{3} (5 x)] + \frac{d}{d x} [64 \sec^{4} (2 x)]$

Step 4.3.4.2

The derivative of $\tan (u_{5})$ with respect to $u_{5}$ is $\sec^{2} (u_{5})$ .

$2625 (- 5 \sin^{3} (5 x) + 10 \sin (5 x) \cos^{2} (5 x)) + 128 (\sec^{2} (2 x) (2 \tan (2 x) (\sec^{2} (u_{5}) \frac{d}{d x} [2 x])) + \tan^{2} (2 x) \frac{d}{d x} [\sec^{2} (2 x)]) + \frac{d}{d x} [- 750 \cos^{3} (5 x)] + \frac{d}{d x} [64 \sec^{4} (2 x)]$

Step 4.3.4.3

Replace all occurrences of $u_{5}$ with $2 x$ .

$2625 (- 5 \sin^{3} (5 x) + 10 \sin (5 x) \cos^{2} (5 x)) + 128 (\sec^{2} (2 x) (2 \tan (2 x) (\sec^{2} (2 x) \frac{d}{d x} [2 x])) + \tan^{2} (2 x) \frac{d}{d x} [\sec^{2} (2 x)]) + \frac{d}{d x} [- 750 \cos^{3} (5 x)] + \frac{d}{d x} [64 \sec^{4} (2 x)]$

Step 4.3.5

Since $2$ is constant with respect to $x$ , the derivative of $2 x$ with respect to $x$ is $2 \frac{d}{d x} [x]$ .

$2625 (- 5 \sin^{3} (5 x) + 10 \sin (5 x) \cos^{2} (5 x)) + 128 (\sec^{2} (2 x) (2 \tan (2 x) (\sec^{2} (2 x) (2 \frac{d}{d x} [x]))) + \tan^{2} (2 x) \frac{d}{d x} [\sec^{2} (2 x)]) + \frac{d}{d x} [- 750 \cos^{3} (5 x)] + \frac{d}{d x} [64 \sec^{4} (2 x)]$

Step 4.3.6

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$2625 (- 5 \sin^{3} (5 x) + 10 \sin (5 x) \cos^{2} (5 x)) + 128 (\sec^{2} (2 x) (2 \tan (2 x) (\sec^{2} (2 x) (2 \cdot 1))) + \tan^{2} (2 x) \frac{d}{d x} [\sec^{2} (2 x)]) + \frac{d}{d x} [- 750 \cos^{3} (5 x)] + \frac{d}{d x} [64 \sec^{4} (2 x)]$

Step 4.3.7

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{2}$ and $g (x) = \sec (2 x)$ .

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Step 4.3.7.1

To apply the Chain Rule, set $u_{6}$ as $\sec (2 x)$ .

$2625 (- 5 \sin^{3} (5 x) + 10 \sin (5 x) \cos^{2} (5 x)) + 128 (\sec^{2} (2 x) (2 \tan (2 x) (\sec^{2} (2 x) (2 \cdot 1))) + \tan^{2} (2 x) (\frac{d}{d u_{6}} [{u_{6}}^{2}] \frac{d}{d x} [\sec (2 x)])) + \frac{d}{d x} [- 750 \cos^{3} (5 x)] + \frac{d}{d x} [64 \sec^{4} (2 x)]$

Step 4.3.7.2

Differentiate using the Power Rule which states that $\frac{d}{d u_{6}} [{u_{6}}^{n}]$ is $n {u_{6}}^{n - 1}$ where $n = 2$ .

$2625 (- 5 \sin^{3} (5 x) + 10 \sin (5 x) \cos^{2} (5 x)) + 128 (\sec^{2} (2 x) (2 \tan (2 x) (\sec^{2} (2 x) (2 \cdot 1))) + \tan^{2} (2 x) (2 u_{6} \frac{d}{d x} [\sec (2 x)])) + \frac{d}{d x} [- 750 \cos^{3} (5 x)] + \frac{d}{d x} [64 \sec^{4} (2 x)]$

Step 4.3.7.3

Replace all occurrences of $u_{6}$ with $\sec (2 x)$ .

$2625 (- 5 \sin^{3} (5 x) + 10 \sin (5 x) \cos^{2} (5 x)) + 128 (\sec^{2} (2 x) (2 \tan (2 x) (\sec^{2} (2 x) (2 \cdot 1))) + \tan^{2} (2 x) (2 \sec (2 x) \frac{d}{d x} [\sec (2 x)])) + \frac{d}{d x} [- 750 \cos^{3} (5 x)] + \frac{d}{d x} [64 \sec^{4} (2 x)]$

Step 4.3.8

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \sec (x)$ and $g (x) = 2 x$ .

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Step 4.3.8.1

To apply the Chain Rule, set $u_{7}$ as $2 x$ .

$2625 (- 5 \sin^{3} (5 x) + 10 \sin (5 x) \cos^{2} (5 x)) + 128 (\sec^{2} (2 x) (2 \tan (2 x) (\sec^{2} (2 x) (2 \cdot 1))) + \tan^{2} (2 x) (2 \sec (2 x) (\frac{d}{d u_{7}} [\sec (u_{7})] \frac{d}{d x} [2 x]))) + \frac{d}{d x} [- 750 \cos^{3} (5 x)] + \frac{d}{d x} [64 \sec^{4} (2 x)]$

Step 4.3.8.2

The derivative of $\sec (u_{7})$ with respect to $u_{7}$ is $\sec (u_{7}) \tan (u_{7})$ .

$2625 (- 5 \sin^{3} (5 x) + 10 \sin (5 x) \cos^{2} (5 x)) + 128 (\sec^{2} (2 x) (2 \tan (2 x) (\sec^{2} (2 x) (2 \cdot 1))) + \tan^{2} (2 x) (2 \sec (2 x) (\sec (u_{7}) \tan (u_{7}) \frac{d}{d x} [2 x]))) + \frac{d}{d x} [- 750 \cos^{3} (5 x)] + \frac{d}{d x} [64 \sec^{4} (2 x)]$

Step 4.3.8.3

Replace all occurrences of $u_{7}$ with $2 x$ .

$2625 (- 5 \sin^{3} (5 x) + 10 \sin (5 x) \cos^{2} (5 x)) + 128 (\sec^{2} (2 x) (2 \tan (2 x) (\sec^{2} (2 x) (2 \cdot 1))) + \tan^{2} (2 x) (2 \sec (2 x) (\sec (2 x) \tan (2 x) \frac{d}{d x} [2 x]))) + \frac{d}{d x} [- 750 \cos^{3} (5 x)] + \frac{d}{d x} [64 \sec^{4} (2 x)]$

Step 4.3.9

Since $2$ is constant with respect to $x$ , the derivative of $2 x$ with respect to $x$ is $2 \frac{d}{d x} [x]$ .

$2625 (- 5 \sin^{3} (5 x) + 10 \sin (5 x) \cos^{2} (5 x)) + 128 (\sec^{2} (2 x) (2 \tan (2 x) (\sec^{2} (2 x) (2 \cdot 1))) + \tan^{2} (2 x) (2 \sec (2 x) (\sec (2 x) \tan (2 x) (2 \frac{d}{d x} [x])))) + \frac{d}{d x} [- 750 \cos^{3} (5 x)] + \frac{d}{d x} [64 \sec^{4} (2 x)]$

Step 4.3.10

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$2625 (- 5 \sin^{3} (5 x) + 10 \sin (5 x) \cos^{2} (5 x)) + 128 (\sec^{2} (2 x) (2 \tan (2 x) (\sec^{2} (2 x) (2 \cdot 1))) + \tan^{2} (2 x) (2 \sec (2 x) (\sec (2 x) \tan (2 x) (2 \cdot 1)))) + \frac{d}{d x} [- 750 \cos^{3} (5 x)] + \frac{d}{d x} [64 \sec^{4} (2 x)]$

Step 4.3.11

Multiply $2$ by $1$ .

$2625 (- 5 \sin^{3} (5 x) + 10 \sin (5 x) \cos^{2} (5 x)) + 128 (\sec^{2} (2 x) (2 \tan (2 x) (\sec^{2} (2 x) \cdot 2)) + \tan^{2} (2 x) (2 \sec (2 x) (\sec (2 x) \tan (2 x) (2 \cdot 1)))) + \frac{d}{d x} [- 750 \cos^{3} (5 x)] + \frac{d}{d x} [64 \sec^{4} (2 x)]$

Step 4.3.12

Move $2$ to the left of $\sec^{2} (2 x)$ .

$2625 (- 5 \sin^{3} (5 x) + 10 \sin (5 x) \cos^{2} (5 x)) + 128 (\sec^{2} (2 x) (2 \tan (2 x) (2 \cdot \sec^{2} (2 x))) + \tan^{2} (2 x) (2 \sec (2 x) (\sec (2 x) \tan (2 x) (2 \cdot 1)))) + \frac{d}{d x} [- 750 \cos^{3} (5 x)] + \frac{d}{d x} [64 \sec^{4} (2 x)]$

Step 4.3.13

Multiply $2$ by $2$ .

$2625 (- 5 \sin^{3} (5 x) + 10 \sin (5 x) \cos^{2} (5 x)) + 128 (\sec^{2} (2 x) (4 \tan (2 x) \sec^{2} (2 x)) + \tan^{2} (2 x) (2 \sec (2 x) (\sec (2 x) \tan (2 x) (2 \cdot 1)))) + \frac{d}{d x} [- 750 \cos^{3} (5 x)] + \frac{d}{d x} [64 \sec^{4} (2 x)]$

Step 4.3.14

Multiply $\sec^{2} (2 x)$ by $\sec^{2} (2 x)$ by adding the exponents.

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Step 4.3.14.1

Move $\sec^{2} (2 x)$ .

$2625 (- 5 \sin^{3} (5 x) + 10 \sin (5 x) \cos^{2} (5 x)) + 128 (\sec^{2} (2 x) \sec^{2} (2 x) (4 \tan (2 x)) + \tan^{2} (2 x) (2 \sec (2 x) (\sec (2 x) \tan (2 x) (2 \cdot 1)))) + \frac{d}{d x} [- 750 \cos^{3} (5 x)] + \frac{d}{d x} [64 \sec^{4} (2 x)]$

Step 4.3.14.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$2625 (- 5 \sin^{3} (5 x) + 10 \sin (5 x) \cos^{2} (5 x)) + 128 ({\sec (2 x)}^{2 + 2} (4 \tan (2 x)) + \tan^{2} (2 x) (2 \sec (2 x) (\sec (2 x) \tan (2 x) (2 \cdot 1)))) + \frac{d}{d x} [- 750 \cos^{3} (5 x)] + \frac{d}{d x} [64 \sec^{4} (2 x)]$

Step 4.3.14.3

Add $2$ and $2$ .

$2625 (- 5 \sin^{3} (5 x) + 10 \sin (5 x) \cos^{2} (5 x)) + 128 (\sec^{4} (2 x) (4 \tan (2 x)) + \tan^{2} (2 x) (2 \sec (2 x) (\sec (2 x) \tan (2 x) (2 \cdot 1)))) + \frac{d}{d x} [- 750 \cos^{3} (5 x)] + \frac{d}{d x} [64 \sec^{4} (2 x)]$

Step 4.3.15

Move $4$ to the left of $\sec^{4} (2 x)$ .

$2625 (- 5 \sin^{3} (5 x) + 10 \sin (5 x) \cos^{2} (5 x)) + 128 (4 \cdot \sec^{4} (2 x) \tan (2 x) + \tan^{2} (2 x) (2 \sec (2 x) (\sec (2 x) \tan (2 x) (2 \cdot 1)))) + \frac{d}{d x} [- 750 \cos^{3} (5 x)] + \frac{d}{d x} [64 \sec^{4} (2 x)]$

Step 4.3.16

Multiply $2$ by $1$ .

$2625 (- 5 \sin^{3} (5 x) + 10 \sin (5 x) \cos^{2} (5 x)) + 128 (4 \sec^{4} (2 x) \tan (2 x) + \tan^{2} (2 x) (2 \sec (2 x) (\sec (2 x) \tan (2 x) \cdot 2))) + \frac{d}{d x} [- 750 \cos^{3} (5 x)] + \frac{d}{d x} [64 \sec^{4} (2 x)]$

Step 4.3.17

Move $2$ to the left of $\sec (2 x) \tan (2 x)$ .

$2625 (- 5 \sin^{3} (5 x) + 10 \sin (5 x) \cos^{2} (5 x)) + 128 (4 \sec^{4} (2 x) \tan (2 x) + \tan^{2} (2 x) (2 \sec (2 x) (2 \cdot (\sec (2 x) \tan (2 x))))) + \frac{d}{d x} [- 750 \cos^{3} (5 x)] + \frac{d}{d x} [64 \sec^{4} (2 x)]$

Step 4.3.18

Multiply $2$ by $2$ .

$2625 (- 5 \sin^{3} (5 x) + 10 \sin (5 x) \cos^{2} (5 x)) + 128 (4 \sec^{4} (2 x) \tan (2 x) + \tan^{2} (2 x) (4 \sec (2 x) (\sec (2 x) \tan (2 x)))) + \frac{d}{d x} [- 750 \cos^{3} (5 x)] + \frac{d}{d x} [64 \sec^{4} (2 x)]$

Step 4.3.19

Raise $\sec (2 x)$ to the power of $1$ .

$2625 (- 5 \sin^{3} (5 x) + 10 \sin (5 x) \cos^{2} (5 x)) + 128 (4 \sec^{4} (2 x) \tan (2 x) + \tan^{2} (2 x) (4 (\sec^{1} (2 x) \sec (2 x)) \tan (2 x))) + \frac{d}{d x} [- 750 \cos^{3} (5 x)] + \frac{d}{d x} [64 \sec^{4} (2 x)]$

Step 4.3.20

Raise $\sec (2 x)$ to the power of $1$ .

$2625 (- 5 \sin^{3} (5 x) + 10 \sin (5 x) \cos^{2} (5 x)) + 128 (4 \sec^{4} (2 x) \tan (2 x) + \tan^{2} (2 x) (4 (\sec^{1} (2 x) \sec^{1} (2 x)) \tan (2 x))) + \frac{d}{d x} [- 750 \cos^{3} (5 x)] + \frac{d}{d x} [64 \sec^{4} (2 x)]$

Step 4.3.21

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$2625 (- 5 \sin^{3} (5 x) + 10 \sin (5 x) \cos^{2} (5 x)) + 128 (4 \sec^{4} (2 x) \tan (2 x) + \tan^{2} (2 x) (4 {\sec (2 x)}^{1 + 1} \tan (2 x))) + \frac{d}{d x} [- 750 \cos^{3} (5 x)] + \frac{d}{d x} [64 \sec^{4} (2 x)]$

Step 4.3.22

Add $1$ and $1$ .

$2625 (- 5 \sin^{3} (5 x) + 10 \sin (5 x) \cos^{2} (5 x)) + 128 (4 \sec^{4} (2 x) \tan (2 x) + \tan^{2} (2 x) (4 \sec^{2} (2 x) \tan (2 x))) + \frac{d}{d x} [- 750 \cos^{3} (5 x)] + \frac{d}{d x} [64 \sec^{4} (2 x)]$

Step 4.3.23

Multiply $\tan^{2} (2 x)$ by $\tan (2 x)$ by adding the exponents.

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Step 4.3.23.1

Move $\tan (2 x)$ .

$2625 (- 5 \sin^{3} (5 x) + 10 \sin (5 x) \cos^{2} (5 x)) + 128 (4 \sec^{4} (2 x) \tan (2 x) + \tan (2 x) \tan^{2} (2 x) (4 \sec^{2} (2 x))) + \frac{d}{d x} [- 750 \cos^{3} (5 x)] + \frac{d}{d x} [64 \sec^{4} (2 x)]$

Step 4.3.23.2

Multiply $\tan (2 x)$ by $\tan^{2} (2 x)$ .

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Step 4.3.23.2.1

Raise $\tan (2 x)$ to the power of $1$ .

$2625 (- 5 \sin^{3} (5 x) + 10 \sin (5 x) \cos^{2} (5 x)) + 128 (4 \sec^{4} (2 x) \tan (2 x) + \tan^{1} (2 x) \tan^{2} (2 x) (4 \sec^{2} (2 x))) + \frac{d}{d x} [- 750 \cos^{3} (5 x)] + \frac{d}{d x} [64 \sec^{4} (2 x)]$

Step 4.3.23.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$2625 (- 5 \sin^{3} (5 x) + 10 \sin (5 x) \cos^{2} (5 x)) + 128 (4 \sec^{4} (2 x) \tan (2 x) + {\tan (2 x)}^{1 + 2} (4 \sec^{2} (2 x))) + \frac{d}{d x} [- 750 \cos^{3} (5 x)] + \frac{d}{d x} [64 \sec^{4} (2 x)]$

Step 4.3.23.3

Add $1$ and $2$ .

$2625 (- 5 \sin^{3} (5 x) + 10 \sin (5 x) \cos^{2} (5 x)) + 128 (4 \sec^{4} (2 x) \tan (2 x) + \tan^{3} (2 x) (4 \sec^{2} (2 x))) + \frac{d}{d x} [- 750 \cos^{3} (5 x)] + \frac{d}{d x} [64 \sec^{4} (2 x)]$

Step 4.3.24

Move $4$ to the left of $\tan^{3} (2 x)$ .

$2625 (- 5 \sin^{3} (5 x) + 10 \sin (5 x) \cos^{2} (5 x)) + 128 (4 \sec^{4} (2 x) \tan (2 x) + 4 \tan^{3} (2 x) \sec^{2} (2 x)) + \frac{d}{d x} [- 750 \cos^{3} (5 x)] + \frac{d}{d x} [64 \sec^{4} (2 x)]$

Step 4.4

Evaluate $\frac{d}{d x} [- 750 \cos^{3} (5 x)]$ .

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Step 4.4.1

Since $- 750$ is constant with respect to $x$ , the derivative of $- 750 \cos^{3} (5 x)$ with respect to $x$ is $- 750 \frac{d}{d x} [\cos^{3} (5 x)]$ .

$2625 (- 5 \sin^{3} (5 x) + 10 \sin (5 x) \cos^{2} (5 x)) + 128 (4 \sec^{4} (2 x) \tan (2 x) + 4 \tan^{3} (2 x) \sec^{2} (2 x)) - 750 \frac{d}{d x} [\cos^{3} (5 x)] + \frac{d}{d x} [64 \sec^{4} (2 x)]$

Step 4.4.2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{3}$ and $g (x) = \cos (5 x)$ .

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Step 4.4.2.1

To apply the Chain Rule, set $u_{8}$ as $\cos (5 x)$ .

$2625 (- 5 \sin^{3} (5 x) + 10 \sin (5 x) \cos^{2} (5 x)) + 128 (4 \sec^{4} (2 x) \tan (2 x) + 4 \tan^{3} (2 x) \sec^{2} (2 x)) - 750 (\frac{d}{d u_{8}} [{u_{8}}^{3}] \frac{d}{d x} [\cos (5 x)]) + \frac{d}{d x} [64 \sec^{4} (2 x)]$

Step 4.4.2.2

Differentiate using the Power Rule which states that $\frac{d}{d u_{8}} [{u_{8}}^{n}]$ is $n {u_{8}}^{n - 1}$ where $n = 3$ .

$2625 (- 5 \sin^{3} (5 x) + 10 \sin (5 x) \cos^{2} (5 x)) + 128 (4 \sec^{4} (2 x) \tan (2 x) + 4 \tan^{3} (2 x) \sec^{2} (2 x)) - 750 (3 {u_{8}}^{2} \frac{d}{d x} [\cos (5 x)]) + \frac{d}{d x} [64 \sec^{4} (2 x)]$

Step 4.4.2.3

Replace all occurrences of $u_{8}$ with $\cos (5 x)$ .

$2625 (- 5 \sin^{3} (5 x) + 10 \sin (5 x) \cos^{2} (5 x)) + 128 (4 \sec^{4} (2 x) \tan (2 x) + 4 \tan^{3} (2 x) \sec^{2} (2 x)) - 750 (3 \cos^{2} (5 x) \frac{d}{d x} [\cos (5 x)]) + \frac{d}{d x} [64 \sec^{4} (2 x)]$

Step 4.4.3

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \cos (x)$ and $g (x) = 5 x$ .

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Step 4.4.3.1

To apply the Chain Rule, set $u_{9}$ as $5 x$ .

$2625 (- 5 \sin^{3} (5 x) + 10 \sin (5 x) \cos^{2} (5 x)) + 128 (4 \sec^{4} (2 x) \tan (2 x) + 4 \tan^{3} (2 x) \sec^{2} (2 x)) - 750 (3 \cos^{2} (5 x) (\frac{d}{d u_{9}} [\cos (u_{9})] \frac{d}{d x} [5 x])) + \frac{d}{d x} [64 \sec^{4} (2 x)]$

Step 4.4.3.2

The derivative of $\cos (u_{9})$ with respect to $u_{9}$ is $- \sin (u_{9})$ .

$2625 (- 5 \sin^{3} (5 x) + 10 \sin (5 x) \cos^{2} (5 x)) + 128 (4 \sec^{4} (2 x) \tan (2 x) + 4 \tan^{3} (2 x) \sec^{2} (2 x)) - 750 (3 \cos^{2} (5 x) (- \sin (u_{9}) \frac{d}{d x} [5 x])) + \frac{d}{d x} [64 \sec^{4} (2 x)]$

Step 4.4.3.3

Replace all occurrences of $u_{9}$ with $5 x$ .

$2625 (- 5 \sin^{3} (5 x) + 10 \sin (5 x) \cos^{2} (5 x)) + 128 (4 \sec^{4} (2 x) \tan (2 x) + 4 \tan^{3} (2 x) \sec^{2} (2 x)) - 750 (3 \cos^{2} (5 x) (- \sin (5 x) \frac{d}{d x} [5 x])) + \frac{d}{d x} [64 \sec^{4} (2 x)]$

Step 4.4.4

Since $5$ is constant with respect to $x$ , the derivative of $5 x$ with respect to $x$ is $5 \frac{d}{d x} [x]$ .

$2625 (- 5 \sin^{3} (5 x) + 10 \sin (5 x) \cos^{2} (5 x)) + 128 (4 \sec^{4} (2 x) \tan (2 x) + 4 \tan^{3} (2 x) \sec^{2} (2 x)) - 750 (3 \cos^{2} (5 x) (- \sin (5 x) (5 \frac{d}{d x} [x]))) + \frac{d}{d x} [64 \sec^{4} (2 x)]$

Step 4.4.5

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$2625 (- 5 \sin^{3} (5 x) + 10 \sin (5 x) \cos^{2} (5 x)) + 128 (4 \sec^{4} (2 x) \tan (2 x) + 4 \tan^{3} (2 x) \sec^{2} (2 x)) - 750 (3 \cos^{2} (5 x) (- \sin (5 x) (5 \cdot 1))) + \frac{d}{d x} [64 \sec^{4} (2 x)]$

Step 4.4.6

Multiply $5$ by $1$ .

$2625 (- 5 \sin^{3} (5 x) + 10 \sin (5 x) \cos^{2} (5 x)) + 128 (4 \sec^{4} (2 x) \tan (2 x) + 4 \tan^{3} (2 x) \sec^{2} (2 x)) - 750 (3 \cos^{2} (5 x) (- \sin (5 x) \cdot 5)) + \frac{d}{d x} [64 \sec^{4} (2 x)]$

Step 4.4.7

Multiply $5$ by $- 1$ .

$2625 (- 5 \sin^{3} (5 x) + 10 \sin (5 x) \cos^{2} (5 x)) + 128 (4 \sec^{4} (2 x) \tan (2 x) + 4 \tan^{3} (2 x) \sec^{2} (2 x)) - 750 (3 \cos^{2} (5 x) (- 5 \sin (5 x))) + \frac{d}{d x} [64 \sec^{4} (2 x)]$

Step 4.4.8

Multiply $- 5$ by $3$ .

$2625 (- 5 \sin^{3} (5 x) + 10 \sin (5 x) \cos^{2} (5 x)) + 128 (4 \sec^{4} (2 x) \tan (2 x) + 4 \tan^{3} (2 x) \sec^{2} (2 x)) - 750 (- 15 \cos^{2} (5 x) \sin (5 x)) + \frac{d}{d x} [64 \sec^{4} (2 x)]$

Step 4.4.9

Multiply $- 15$ by $- 750$ .

$2625 (- 5 \sin^{3} (5 x) + 10 \sin (5 x) \cos^{2} (5 x)) + 128 (4 \sec^{4} (2 x) \tan (2 x) + 4 \tan^{3} (2 x) \sec^{2} (2 x)) + 11250 \cos^{2} (5 x) \sin (5 x) + \frac{d}{d x} [64 \sec^{4} (2 x)]$

Step 4.5

Evaluate $\frac{d}{d x} [64 \sec^{4} (2 x)]$ .

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Step 4.5.1

Since $64$ is constant with respect to $x$ , the derivative of $64 \sec^{4} (2 x)$ with respect to $x$ is $64 \frac{d}{d x} [\sec^{4} (2 x)]$ .

$2625 (- 5 \sin^{3} (5 x) + 10 \sin (5 x) \cos^{2} (5 x)) + 128 (4 \sec^{4} (2 x) \tan (2 x) + 4 \tan^{3} (2 x) \sec^{2} (2 x)) + 11250 \cos^{2} (5 x) \sin (5 x) + 64 \frac{d}{d x} [\sec^{4} (2 x)]$

Step 4.5.2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{4}$ and $g (x) = \sec (2 x)$ .

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Step 4.5.2.1

To apply the Chain Rule, set $u_{10}$ as $\sec (2 x)$ .

$2625 (- 5 \sin^{3} (5 x) + 10 \sin (5 x) \cos^{2} (5 x)) + 128 (4 \sec^{4} (2 x) \tan (2 x) + 4 \tan^{3} (2 x) \sec^{2} (2 x)) + 11250 \cos^{2} (5 x) \sin (5 x) + 64 (\frac{d}{d u_{10}} [{u_{10}}^{4}] \frac{d}{d x} [\sec (2 x)])$

Step 4.5.2.2

Differentiate using the Power Rule which states that $\frac{d}{d u_{10}} [{u_{10}}^{n}]$ is $n {u_{10}}^{n - 1}$ where $n = 4$ .

$2625 (- 5 \sin^{3} (5 x) + 10 \sin (5 x) \cos^{2} (5 x)) + 128 (4 \sec^{4} (2 x) \tan (2 x) + 4 \tan^{3} (2 x) \sec^{2} (2 x)) + 11250 \cos^{2} (5 x) \sin (5 x) + 64 (4 {u_{10}}^{3} \frac{d}{d x} [\sec (2 x)])$

Step 4.5.2.3

Replace all occurrences of $u_{10}$ with $\sec (2 x)$ .

$2625 (- 5 \sin^{3} (5 x) + 10 \sin (5 x) \cos^{2} (5 x)) + 128 (4 \sec^{4} (2 x) \tan (2 x) + 4 \tan^{3} (2 x) \sec^{2} (2 x)) + 11250 \cos^{2} (5 x) \sin (5 x) + 64 (4 \sec^{3} (2 x) \frac{d}{d x} [\sec (2 x)])$

Step 4.5.3

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \sec (x)$ and $g (x) = 2 x$ .

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Step 4.5.3.1

To apply the Chain Rule, set $u_{11}$ as $2 x$ .

$2625 (- 5 \sin^{3} (5 x) + 10 \sin (5 x) \cos^{2} (5 x)) + 128 (4 \sec^{4} (2 x) \tan (2 x) + 4 \tan^{3} (2 x) \sec^{2} (2 x)) + 11250 \cos^{2} (5 x) \sin (5 x) + 64 (4 \sec^{3} (2 x) (\frac{d}{d u_{11}} [\sec (u_{11})] \frac{d}{d x} [2 x]))$

Step 4.5.3.2

The derivative of $\sec (u_{11})$ with respect to $u_{11}$ is $\sec (u_{11}) \tan (u_{11})$ .

$2625 (- 5 \sin^{3} (5 x) + 10 \sin (5 x) \cos^{2} (5 x)) + 128 (4 \sec^{4} (2 x) \tan (2 x) + 4 \tan^{3} (2 x) \sec^{2} (2 x)) + 11250 \cos^{2} (5 x) \sin (5 x) + 64 (4 \sec^{3} (2 x) (\sec (u_{11}) \tan (u_{11}) \frac{d}{d x} [2 x]))$

Step 4.5.3.3

Replace all occurrences of $u_{11}$ with $2 x$ .

$2625 (- 5 \sin^{3} (5 x) + 10 \sin (5 x) \cos^{2} (5 x)) + 128 (4 \sec^{4} (2 x) \tan (2 x) + 4 \tan^{3} (2 x) \sec^{2} (2 x)) + 11250 \cos^{2} (5 x) \sin (5 x) + 64 (4 \sec^{3} (2 x) (\sec (2 x) \tan (2 x) \frac{d}{d x} [2 x]))$

Step 4.5.4

Since $2$ is constant with respect to $x$ , the derivative of $2 x$ with respect to $x$ is $2 \frac{d}{d x} [x]$ .

Step 4.5.5

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

Step 4.5.6

Multiply $2$ by $1$ .

Step 4.5.7

Move $2$ to the left of $\sec (2 x) \tan (2 x)$ .

$2625 (- 5 \sin^{3} (5 x) + 10 \sin (5 x) \cos^{2} (5 x)) + 128 (4 \sec^{4} (2 x) \tan (2 x) + 4 \tan^{3} (2 x) \sec^{2} (2 x)) + 11250 \cos^{2} (5 x) \sin (5 x) + 64 (4 \sec^{3} (2 x) (2 \cdot (\sec (2 x) \tan (2 x))))$

Step 4.5.8

Multiply $2$ by $4$ .

$2625 (- 5 \sin^{3} (5 x) + 10 \sin (5 x) \cos^{2} (5 x)) + 128 (4 \sec^{4} (2 x) \tan (2 x) + 4 \tan^{3} (2 x) \sec^{2} (2 x)) + 11250 \cos^{2} (5 x) \sin (5 x) + 64 (8 \sec^{3} (2 x) (\sec (2 x) \tan (2 x)))$

Step 4.5.9

Raise $\sec (2 x)$ to the power of $1$ .

$2625 (- 5 \sin^{3} (5 x) + 10 \sin (5 x) \cos^{2} (5 x)) + 128 (4 \sec^{4} (2 x) \tan (2 x) + 4 \tan^{3} (2 x) \sec^{2} (2 x)) + 11250 \cos^{2} (5 x) \sin (5 x) + 64 (8 (\sec^{1} (2 x) \sec^{3} (2 x)) \tan (2 x))$

Step 4.5.10

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$2625 (- 5 \sin^{3} (5 x) + 10 \sin (5 x) \cos^{2} (5 x)) + 128 (4 \sec^{4} (2 x) \tan (2 x) + 4 \tan^{3} (2 x) \sec^{2} (2 x)) + 11250 \cos^{2} (5 x) \sin (5 x) + 64 (8 {\sec (2 x)}^{1 + 3} \tan (2 x))$

Step 4.5.11

Add $1$ and $3$ .

$2625 (- 5 \sin^{3} (5 x) + 10 \sin (5 x) \cos^{2} (5 x)) + 128 (4 \sec^{4} (2 x) \tan (2 x) + 4 \tan^{3} (2 x) \sec^{2} (2 x)) + 11250 \cos^{2} (5 x) \sin (5 x) + 64 (8 \sec^{4} (2 x) \tan (2 x))$

Step 4.5.12

Multiply $8$ by $64$ .

$2625 (- 5 \sin^{3} (5 x) + 10 \sin (5 x) \cos^{2} (5 x)) + 128 (4 \sec^{4} (2 x) \tan (2 x) + 4 \tan^{3} (2 x) \sec^{2} (2 x)) + 11250 \cos^{2} (5 x) \sin (5 x) + 512 \sec^{4} (2 x) \tan (2 x)$

Step 4.6

Simplify.

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Step 4.6.1

Apply the distributive property.

$2625 (- 5 \sin^{3} (5 x)) + 2625 (10 \sin (5 x) \cos^{2} (5 x)) + 128 (4 \sec^{4} (2 x) \tan (2 x) + 4 \tan^{3} (2 x) \sec^{2} (2 x)) + 11250 \cos^{2} (5 x) \sin (5 x) + 512 \sec^{4} (2 x) \tan (2 x)$

Step 4.6.2

Apply the distributive property.

$2625 (- 5 \sin^{3} (5 x)) + 2625 (10 \sin (5 x) \cos^{2} (5 x)) + 128 (4 \sec^{4} (2 x) \tan (2 x)) + 128 (4 \tan^{3} (2 x) \sec^{2} (2 x)) + 11250 \cos^{2} (5 x) \sin (5 x) + 512 \sec^{4} (2 x) \tan (2 x)$

Step 4.6.3

Combine terms.

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Step 4.6.3.1

Multiply $- 5$ by $2625$ .

$- 13125 \sin^{3} (5 x) + 2625 (10 \sin (5 x) \cos^{2} (5 x)) + 128 (4 \sec^{4} (2 x) \tan (2 x)) + 128 (4 \tan^{3} (2 x) \sec^{2} (2 x)) + 11250 \cos^{2} (5 x) \sin (5 x) + 512 \sec^{4} (2 x) \tan (2 x)$

Step 4.6.3.2

Multiply $10$ by $2625$ .

$- 13125 \sin^{3} (5 x) + 26250 (\sin (5 x) \cos^{2} (5 x)) + 128 (4 \sec^{4} (2 x) \tan (2 x)) + 128 (4 \tan^{3} (2 x) \sec^{2} (2 x)) + 11250 \cos^{2} (5 x) \sin (5 x) + 512 \sec^{4} (2 x) \tan (2 x)$

Step 4.6.3.3

Multiply $4$ by $128$ .

$- 13125 \sin^{3} (5 x) + 26250 \sin (5 x) \cos^{2} (5 x) + 512 (\sec^{4} (2 x) \tan (2 x)) + 128 (4 \tan^{3} (2 x) \sec^{2} (2 x)) + 11250 \cos^{2} (5 x) \sin (5 x) + 512 \sec^{4} (2 x) \tan (2 x)$

Step 4.6.3.4

Multiply $4$ by $128$ .

$- 13125 \sin^{3} (5 x) + 26250 \sin (5 x) \cos^{2} (5 x) + 512 \sec^{4} (2 x) \tan (2 x) + 512 (\tan^{3} (2 x) \sec^{2} (2 x)) + 11250 \cos^{2} (5 x) \sin (5 x) + 512 \sec^{4} (2 x) \tan (2 x)$

Step 4.6.3.5

Reorder the factors of $26250 \sin (5 x) \cos^{2} (5 x)$ .

$- 13125 \sin^{3} (5 x) + 26250 \cos^{2} (5 x) \sin (5 x) + 512 \sec^{4} (2 x) \tan (2 x) + 512 \tan^{3} (2 x) \sec^{2} (2 x) + 11250 \cos^{2} (5 x) \sin (5 x) + 512 \sec^{4} (2 x) \tan (2 x)$

Step 4.6.3.6

Add $26250 \cos^{2} (5 x) \sin (5 x)$ and $11250 \cos^{2} (5 x) \sin (5 x)$ .

$- 13125 \sin^{3} (5 x) + 37500 \cos^{2} (5 x) \sin (5 x) + 512 \sec^{4} (2 x) \tan (2 x) + 512 \tan^{3} (2 x) \sec^{2} (2 x) + 512 \sec^{4} (2 x) \tan (2 x)$

Step 4.6.3.7

Add $512 \sec^{4} (2 x) \tan (2 x)$ and $512 \sec^{4} (2 x) \tan (2 x)$ .

$- 13125 \sin^{3} (5 x) + 37500 \cos^{2} (5 x) \sin (5 x) + 1024 \sec^{4} (2 x) \tan (2 x) + 512 \tan^{3} (2 x) \sec^{2} (2 x)$

Step 4.6.4

Reorder terms.

$37500 \cos^{2} (5 x) \sin (5 x) + 1024 \sec^{4} (2 x) \tan (2 x) + 512 \tan^{3} (2 x) \sec^{2} (2 x) - 13125 \sin^{3} (5 x)$

Step 4.6.5

Simplify each term.

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Step 4.6.5.1

Rewrite $\sec (2 x)$ in terms of sines and cosines.

$37500 \cos^{2} (5 x) \sin (5 x) + 1024 {(\frac{1}{\cos (2 x)})}^{4} \tan (2 x) + 512 \tan^{3} (2 x) \sec^{2} (2 x) - 13125 \sin^{3} (5 x)$

Step 4.6.5.2

Apply the product rule to $\frac{1}{\cos (2 x)}$ .

$37500 \cos^{2} (5 x) \sin (5 x) + 1024 \frac{1^{4}}{\cos^{4} (2 x)} \tan (2 x) + 512 \tan^{3} (2 x) \sec^{2} (2 x) - 13125 \sin^{3} (5 x)$

Step 4.6.5.3

One to any power is one.

$37500 \cos^{2} (5 x) \sin (5 x) + 1024 \frac{1}{\cos^{4} (2 x)} \tan (2 x) + 512 \tan^{3} (2 x) \sec^{2} (2 x) - 13125 \sin^{3} (5 x)$

Step 4.6.5.4

Combine $1024$ and $\frac{1}{\cos^{4} (2 x)}$ .

$37500 \cos^{2} (5 x) \sin (5 x) + \frac{1024}{\cos^{4} (2 x)} \tan (2 x) + 512 \tan^{3} (2 x) \sec^{2} (2 x) - 13125 \sin^{3} (5 x)$

Step 4.6.5.5

Rewrite $\tan (2 x)$ in terms of sines and cosines.

$37500 \cos^{2} (5 x) \sin (5 x) + \frac{1024}{\cos^{4} (2 x)} \cdot \frac{\sin (2 x)}{\cos (2 x)} + 512 \tan^{3} (2 x) \sec^{2} (2 x) - 13125 \sin^{3} (5 x)$

Step 4.6.5.6

Combine.

$37500 \cos^{2} (5 x) \sin (5 x) + \frac{1024 \sin (2 x)}{\cos^{4} (2 x) \cos (2 x)} + 512 \tan^{3} (2 x) \sec^{2} (2 x) - 13125 \sin^{3} (5 x)$

Step 4.6.5.7

Multiply $\cos^{4} (2 x)$ by $\cos (2 x)$ by adding the exponents.

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Step 4.6.5.7.1

Multiply $\cos^{4} (2 x)$ by $\cos (2 x)$ .

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Step 4.6.5.7.1.1

Raise $\cos (2 x)$ to the power of $1$ .

$37500 \cos^{2} (5 x) \sin (5 x) + \frac{1024 \sin (2 x)}{\cos^{4} (2 x) \cos^{1} (2 x)} + 512 \tan^{3} (2 x) \sec^{2} (2 x) - 13125 \sin^{3} (5 x)$

Step 4.6.5.7.1.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$37500 \cos^{2} (5 x) \sin (5 x) + \frac{1024 \sin (2 x)}{{\cos (2 x)}^{4 + 1}} + 512 \tan^{3} (2 x) \sec^{2} (2 x) - 13125 \sin^{3} (5 x)$

Step 4.6.5.7.2

Add $4$ and $1$ .

$37500 \cos^{2} (5 x) \sin (5 x) + \frac{1024 \sin (2 x)}{\cos^{5} (2 x)} + 512 \tan^{3} (2 x) \sec^{2} (2 x) - 13125 \sin^{3} (5 x)$

Step 4.6.5.8

Rewrite $\tan (2 x)$ in terms of sines and cosines.

$37500 \cos^{2} (5 x) \sin (5 x) + \frac{1024 \sin (2 x)}{\cos^{5} (2 x)} + 512 {(\frac{\sin (2 x)}{\cos (2 x)})}^{3} \sec^{2} (2 x) - 13125 \sin^{3} (5 x)$

Step 4.6.5.9

Apply the product rule to $\frac{\sin (2 x)}{\cos (2 x)}$ .

$37500 \cos^{2} (5 x) \sin (5 x) + \frac{1024 \sin (2 x)}{\cos^{5} (2 x)} + 512 \frac{\sin^{3} (2 x)}{\cos^{3} (2 x)} \sec^{2} (2 x) - 13125 \sin^{3} (5 x)$

Step 4.6.5.10

Combine $512$ and $\frac{\sin^{3} (2 x)}{\cos^{3} (2 x)}$ .

$37500 \cos^{2} (5 x) \sin (5 x) + \frac{1024 \sin (2 x)}{\cos^{5} (2 x)} + \frac{512 \sin^{3} (2 x)}{\cos^{3} (2 x)} \sec^{2} (2 x) - 13125 \sin^{3} (5 x)$

Step 4.6.5.11

Rewrite $\sec (2 x)$ in terms of sines and cosines.

$37500 \cos^{2} (5 x) \sin (5 x) + \frac{1024 \sin (2 x)}{\cos^{5} (2 x)} + \frac{512 \sin^{3} (2 x)}{\cos^{3} (2 x)} {(\frac{1}{\cos (2 x)})}^{2} - 13125 \sin^{3} (5 x)$

Step 4.6.5.12

Apply the product rule to $\frac{1}{\cos (2 x)}$ .

$37500 \cos^{2} (5 x) \sin (5 x) + \frac{1024 \sin (2 x)}{\cos^{5} (2 x)} + \frac{512 \sin^{3} (2 x)}{\cos^{3} (2 x)} \cdot \frac{1^{2}}{\cos^{2} (2 x)} - 13125 \sin^{3} (5 x)$

Step 4.6.5.13

Combine.

$37500 \cos^{2} (5 x) \sin (5 x) + \frac{1024 \sin (2 x)}{\cos^{5} (2 x)} + \frac{512 \sin^{3} (2 x) \cdot 1^{2}}{\cos^{3} (2 x) \cos^{2} (2 x)} - 13125 \sin^{3} (5 x)$

Step 4.6.5.14

Multiply $\cos^{3} (2 x)$ by $\cos^{2} (2 x)$ by adding the exponents.

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Step 4.6.5.14.1

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$37500 \cos^{2} (5 x) \sin (5 x) + \frac{1024 \sin (2 x)}{\cos^{5} (2 x)} + \frac{512 \sin^{3} (2 x) \cdot 1^{2}}{{\cos (2 x)}^{3 + 2}} - 13125 \sin^{3} (5 x)$

Step 4.6.5.14.2

Add $3$ and $2$ .

$37500 \cos^{2} (5 x) \sin (5 x) + \frac{1024 \sin (2 x)}{\cos^{5} (2 x)} + \frac{512 \sin^{3} (2 x) \cdot 1^{2}}{\cos^{5} (2 x)} - 13125 \sin^{3} (5 x)$

Step 4.6.5.15

Simplify the numerator.

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Step 4.6.5.15.1

One to any power is one.

$37500 \cos^{2} (5 x) \sin (5 x) + \frac{1024 \sin (2 x)}{\cos^{5} (2 x)} + \frac{512 \sin^{3} (2 x) \cdot 1}{\cos^{5} (2 x)} - 13125 \sin^{3} (5 x)$

Step 4.6.5.15.2

Multiply $512$ by $1$ .

$37500 \cos^{2} (5 x) \sin (5 x) + \frac{1024 \sin (2 x)}{\cos^{5} (2 x)} + \frac{512 \sin^{3} (2 x)}{\cos^{5} (2 x)} - 13125 \sin^{3} (5 x)$

Step 4.6.6

Simplify each term.

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Step 4.6.6.1

Factor $\cos (2 x)$ out of $\cos^{5} (2 x)$ .

$37500 \cos^{2} (5 x) \sin (5 x) + \frac{1024 \sin (2 x)}{\cos (2 x) \cos^{4} (2 x)} + \frac{512 \sin^{3} (2 x)}{\cos^{5} (2 x)} - 13125 \sin^{3} (5 x)$

Step 4.6.6.2

Separate fractions.

$37500 \cos^{2} (5 x) \sin (5 x) + \frac{1024}{\cos^{4} (2 x)} \cdot \frac{\sin (2 x)}{\cos (2 x)} + \frac{512 \sin^{3} (2 x)}{\cos^{5} (2 x)} - 13125 \sin^{3} (5 x)$

Step 4.6.6.3

Convert from $\frac{\sin (2 x)}{\cos (2 x)}$ to $\tan (2 x)$ .

$37500 \cos^{2} (5 x) \sin (5 x) + \frac{1024}{\cos^{4} (2 x)} \tan (2 x) + \frac{512 \sin^{3} (2 x)}{\cos^{5} (2 x)} - 13125 \sin^{3} (5 x)$

Step 4.6.6.4

Multiply by $1$ .

$37500 \cos^{2} (5 x) \sin (5 x) + \frac{1024}{\cos^{4} (2 x) \cdot 1} \tan (2 x) + \frac{512 \sin^{3} (2 x)}{\cos^{5} (2 x)} - 13125 \sin^{3} (5 x)$

Step 4.6.6.5

Separate fractions.

$37500 \cos^{2} (5 x) \sin (5 x) + \frac{1024}{1} \cdot \frac{1}{\cos^{4} (2 x)} \tan (2 x) + \frac{512 \sin^{3} (2 x)}{\cos^{5} (2 x)} - 13125 \sin^{3} (5 x)$

Step 4.6.6.6

Convert from $\frac{1}{\cos^{4} (2 x)}$ to $\sec^{4} (2 x)$ .

$37500 \cos^{2} (5 x) \sin (5 x) + \frac{1024}{1} \sec^{4} (2 x) \tan (2 x) + \frac{512 \sin^{3} (2 x)}{\cos^{5} (2 x)} - 13125 \sin^{3} (5 x)$

Step 4.6.6.7

Divide $1024$ by $1$ .

$37500 \cos^{2} (5 x) \sin (5 x) + 1024 \sec^{4} (2 x) \tan (2 x) + \frac{512 \sin^{3} (2 x)}{\cos^{5} (2 x)} - 13125 \sin^{3} (5 x)$

Step 4.6.6.8

Multiply by $1$ .

$37500 \cos^{2} (5 x) \sin (5 x) + 1024 \sec^{4} (2 x) \tan (2 x) + \frac{512 (\sin^{3} (2 x) \cdot 1)}{\cos^{5} (2 x)} - 13125 \sin^{3} (5 x)$

Step 4.6.6.9

Factor $\cos^{3} (2 x)$ out of $\cos^{5} (2 x)$ .

$37500 \cos^{2} (5 x) \sin (5 x) + 1024 \sec^{4} (2 x) \tan (2 x) + \frac{512 (\sin^{3} (2 x) \cdot 1)}{\cos^{3} (2 x) \cos^{2} (2 x)} - 13125 \sin^{3} (5 x)$

Step 4.6.6.10

Separate fractions.

$37500 \cos^{2} (5 x) \sin (5 x) + 1024 \sec^{4} (2 x) \tan (2 x) + \frac{512 \cdot (1)}{\cos^{2} (2 x)} \cdot \frac{\sin^{3} (2 x)}{\cos^{3} (2 x)} - 13125 \sin^{3} (5 x)$

Step 4.6.6.11

Convert from $\frac{\sin^{3} (2 x)}{\cos^{3} (2 x)}$ to $\tan^{3} (2 x)$ .

$37500 \cos^{2} (5 x) \sin (5 x) + 1024 \sec^{4} (2 x) \tan (2 x) + \frac{512 \cdot (1)}{\cos^{2} (2 x)} \tan^{3} (2 x) - 13125 \sin^{3} (5 x)$

Step 4.6.6.12

Multiply $512$ by $1$ .

$37500 \cos^{2} (5 x) \sin (5 x) + 1024 \sec^{4} (2 x) \tan (2 x) + \frac{512}{\cos^{2} (2 x)} \tan^{3} (2 x) - 13125 \sin^{3} (5 x)$

Step 4.6.6.13

Multiply by $1$ .

$37500 \cos^{2} (5 x) \sin (5 x) + 1024 \sec^{4} (2 x) \tan (2 x) + \frac{512}{\cos^{2} (2 x) \cdot 1} \tan^{3} (2 x) - 13125 \sin^{3} (5 x)$

Step 4.6.6.14

Separate fractions.

$37500 \cos^{2} (5 x) \sin (5 x) + 1024 \sec^{4} (2 x) \tan (2 x) + \frac{512}{1} \cdot \frac{1}{\cos^{2} (2 x)} \tan^{3} (2 x) - 13125 \sin^{3} (5 x)$

Step 4.6.6.15

Convert from $\frac{1}{\cos^{2} (2 x)}$ to $\sec^{2} (2 x)$ .

$37500 \cos^{2} (5 x) \sin (5 x) + 1024 \sec^{4} (2 x) \tan (2 x) + \frac{512}{1} \sec^{2} (2 x) \tan^{3} (2 x) - 13125 \sin^{3} (5 x)$

Step 4.6.6.16

Divide $512$ by $1$ .

$f^{4} (x) = 37500 \cos^{2} (5 x) \sin (5 x) + 1024 \sec^{4} (2 x) \tan (2 x) + 512 \sec^{2} (2 x) \tan^{3} (2 x) - 13125 \sin^{3} (5 x)$