Calculus Examples

$y = \cos (x) + \tan (x)$

Step 1

Find the first derivative.

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Step 1.1

By the Sum Rule, the derivative of $\cos (x) + \tan (x)$ with respect to $x$ is $\frac{d}{d x} [\cos (x)] + \frac{d}{d x} [\tan (x)]$ .

$\frac{d}{d x} [\cos (x)] + \frac{d}{d x} [\tan (x)]$

Step 1.2

The derivative of $\cos (x)$ with respect to $x$ is $- \sin (x)$ .

$- \sin (x) + \frac{d}{d x} [\tan (x)]$

Step 1.3

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

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

Step 2

Find the second derivative.

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Step 2.1

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

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

Step 2.2

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

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Step 2.2.1

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

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

Step 2.2.2

The derivative of $\sin (x)$ with respect to $x$ is $\cos (x)$ .

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

Step 2.3

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

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Step 2.3.1

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 $g (x) = \sec (x)$ .

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Step 2.3.1.1

To apply the Chain Rule, set $u$ as $\sec (x)$ .

$- \cos (x) + \frac{d}{d u} [u^{2}] \frac{d}{d x} [\sec (x)]$

Step 2.3.1.2

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

$- \cos (x) + 2 u \frac{d}{d x} [\sec (x)]$

Step 2.3.1.3

Replace all occurrences of $u$ with $\sec (x)$ .

$- \cos (x) + 2 \sec (x) \frac{d}{d x} [\sec (x)]$

Step 2.3.2

The derivative of $\sec (x)$ with respect to $x$ is $\sec (x) \tan (x)$ .

$- \cos (x) + 2 \sec (x) (\sec (x) \tan (x))$

Step 2.3.3

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

$- \cos (x) + 2 (\sec^{1} (x) \sec (x)) \tan (x)$

Step 2.3.4

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

$- \cos (x) + 2 (\sec^{1} (x) \sec^{1} (x)) \tan (x)$

Step 2.3.5

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

$- \cos (x) + 2 {\sec (x)}^{1 + 1} \tan (x)$

Step 2.3.6

Add $1$ and $1$ .

$- \cos (x) + 2 \sec^{2} (x) \tan (x)$

Step 2.4

Simplify.

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Step 2.4.1

Reorder terms.

$2 \sec^{2} (x) \tan (x) - \cos (x)$

Step 2.4.2

Simplify each term.

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Step 2.4.2.1

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

$2 {(\frac{1}{\cos (x)})}^{2} \tan (x) - \cos (x)$

Step 2.4.2.2

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

$2 \frac{1^{2}}{\cos^{2} (x)} \tan (x) - \cos (x)$

Step 2.4.2.3

One to any power is one.

$2 \frac{1}{\cos^{2} (x)} \tan (x) - \cos (x)$

Step 2.4.2.4

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

$\frac{2}{\cos^{2} (x)} \tan (x) - \cos (x)$

Step 2.4.2.5

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

$\frac{2}{\cos^{2} (x)} \cdot \frac{\sin (x)}{\cos (x)} - \cos (x)$

Step 2.4.2.6

Combine.

$\frac{2 \sin (x)}{\cos^{2} (x) \cos (x)} - \cos (x)$

Step 2.4.2.7

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

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Step 2.4.2.7.1

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

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Step 2.4.2.7.1.1

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

$\frac{2 \sin (x)}{\cos^{2} (x) \cos^{1} (x)} - \cos (x)$

Step 2.4.2.7.1.2

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

$\frac{2 \sin (x)}{{\cos (x)}^{2 + 1}} - \cos (x)$

Step 2.4.2.7.2

Add $2$ and $1$ .

$\frac{2 \sin (x)}{\cos^{3} (x)} - \cos (x)$

Step 2.4.3

Simplify each term.

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Step 2.4.3.1

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

$\frac{2 \sin (x)}{\cos (x) \cos^{2} (x)} - \cos (x)$

Step 2.4.3.2

Separate fractions.

$\frac{2}{\cos^{2} (x)} \cdot \frac{\sin (x)}{\cos (x)} - \cos (x)$

Step 2.4.3.3

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

$\frac{2}{\cos^{2} (x)} \tan (x) - \cos (x)$

Step 2.4.3.4

Multiply by $1$ .

$\frac{2}{\cos^{2} (x) \cdot 1} \tan (x) - \cos (x)$

Step 2.4.3.5

Separate fractions.

$\frac{2}{1} \cdot \frac{1}{\cos^{2} (x)} \tan (x) - \cos (x)$

Step 2.4.3.6

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

$\frac{2}{1} \sec^{2} (x) \tan (x) - \cos (x)$

Step 2.4.3.7

Divide $2$ by $1$ .

$f'' (x) = 2 \sec^{2} (x) \tan (x) - \cos (x)$

Step 3

Find the third derivative.

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Step 3.1

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

$\frac{d}{d x} [2 \sec^{2} (x) \tan (x)] + \frac{d}{d x} [- \cos (x)]$

Step 3.2

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

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Step 3.2.1

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

$2 \frac{d}{d x} [\sec^{2} (x) \tan (x)] + \frac{d}{d x} [- \cos (x)]$

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 $f (x) = \sec^{2} (x)$ and $g (x) = \tan (x)$ .

$2 (\sec^{2} (x) \frac{d}{d x} [\tan (x)] + \tan (x) \frac{d}{d x} [\sec^{2} (x)]) + \frac{d}{d x} [- \cos (x)]$

Step 3.2.3

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

$2 (\sec^{2} (x) \sec^{2} (x) + \tan (x) \frac{d}{d x} [\sec^{2} (x)]) + \frac{d}{d x} [- \cos (x)]$

Step 3.2.4

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

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Step 3.2.4.1

To apply the Chain Rule, set $u$ as $\sec (x)$ .

$2 (\sec^{2} (x) \sec^{2} (x) + \tan (x) (\frac{d}{d u} [u^{2}] \frac{d}{d x} [\sec (x)])) + \frac{d}{d x} [- \cos (x)]$

Step 3.2.4.2

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

$2 (\sec^{2} (x) \sec^{2} (x) + \tan (x) (2 u \frac{d}{d x} [\sec (x)])) + \frac{d}{d x} [- \cos (x)]$

Step 3.2.4.3

Replace all occurrences of $u$ with $\sec (x)$ .

$2 (\sec^{2} (x) \sec^{2} (x) + \tan (x) (2 \sec (x) \frac{d}{d x} [\sec (x)])) + \frac{d}{d x} [- \cos (x)]$

Step 3.2.5

The derivative of $\sec (x)$ with respect to $x$ is $\sec (x) \tan (x)$ .

$2 (\sec^{2} (x) \sec^{2} (x) + \tan (x) (2 \sec (x) (\sec (x) \tan (x)))) + \frac{d}{d x} [- \cos (x)]$

Step 3.2.6

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

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Step 3.2.6.1

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

$2 ({\sec (x)}^{2 + 2} + \tan (x) (2 \sec (x) (\sec (x) \tan (x)))) + \frac{d}{d x} [- \cos (x)]$

Step 3.2.6.2

Add $2$ and $2$ .

$2 (\sec^{4} (x) + \tan (x) (2 \sec (x) (\sec (x) \tan (x)))) + \frac{d}{d x} [- \cos (x)]$

Step 3.2.7

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

$2 (\sec^{4} (x) + \tan (x) (2 (\sec^{1} (x) \sec (x)) \tan (x))) + \frac{d}{d x} [- \cos (x)]$

Step 3.2.8

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

$2 (\sec^{4} (x) + \tan (x) (2 (\sec^{1} (x) \sec^{1} (x)) \tan (x))) + \frac{d}{d x} [- \cos (x)]$

Step 3.2.9

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

$2 (\sec^{4} (x) + \tan (x) (2 {\sec (x)}^{1 + 1} \tan (x))) + \frac{d}{d x} [- \cos (x)]$

Step 3.2.10

Add $1$ and $1$ .

$2 (\sec^{4} (x) + \tan (x) (2 \sec^{2} (x) \tan (x))) + \frac{d}{d x} [- \cos (x)]$

Step 3.2.11

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

$2 (\sec^{4} (x) + 2 \sec^{2} (x) (\tan^{1} (x) \tan (x))) + \frac{d}{d x} [- \cos (x)]$

Step 3.2.12

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

$2 (\sec^{4} (x) + 2 \sec^{2} (x) (\tan^{1} (x) \tan^{1} (x))) + \frac{d}{d x} [- \cos (x)]$

Step 3.2.13

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

$2 (\sec^{4} (x) + 2 \sec^{2} (x) {\tan (x)}^{1 + 1}) + \frac{d}{d x} [- \cos (x)]$

Step 3.2.14

Add $1$ and $1$ .

$2 (\sec^{4} (x) + 2 \sec^{2} (x) \tan^{2} (x)) + \frac{d}{d x} [- \cos (x)]$

Step 3.3

Evaluate $\frac{d}{d x} [- \cos (x)]$ .

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Step 3.3.1

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

$2 (\sec^{4} (x) + 2 \sec^{2} (x) \tan^{2} (x)) - \frac{d}{d x} [\cos (x)]$

Step 3.3.2

The derivative of $\cos (x)$ with respect to $x$ is $- \sin (x)$ .

$2 (\sec^{4} (x) + 2 \sec^{2} (x) \tan^{2} (x)) - - \sin (x)$

Step 3.3.3

Multiply $- 1$ by $- 1$ .

$2 (\sec^{4} (x) + 2 \sec^{2} (x) \tan^{2} (x)) + 1 \sin (x)$

Step 3.3.4

Multiply $\sin (x)$ by $1$ .

$2 (\sec^{4} (x) + 2 \sec^{2} (x) \tan^{2} (x)) + \sin (x)$

Step 3.4

Simplify.

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Step 3.4.1

Apply the distributive property.

$2 \sec^{4} (x) + 2 (2 \sec^{2} (x) \tan^{2} (x)) + \sin (x)$

Step 3.4.2

Multiply $2$ by $2$ .

$2 \sec^{4} (x) + 4 \sec^{2} (x) \tan^{2} (x) + \sin (x)$

Step 3.4.3

Reorder terms.

$4 \sec^{2} (x) \tan^{2} (x) + 2 \sec^{4} (x) + \sin (x)$

Step 3.4.4

Simplify each term.

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Step 3.4.4.1

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

$4 {(\frac{1}{\cos (x)})}^{2} \tan^{2} (x) + 2 \sec^{4} (x) + \sin (x)$

Step 3.4.4.2

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

$4 \frac{1^{2}}{\cos^{2} (x)} \tan^{2} (x) + 2 \sec^{4} (x) + \sin (x)$

Step 3.4.4.3

One to any power is one.

$4 \frac{1}{\cos^{2} (x)} \tan^{2} (x) + 2 \sec^{4} (x) + \sin (x)$

Step 3.4.4.4

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

$\frac{4}{\cos^{2} (x)} \tan^{2} (x) + 2 \sec^{4} (x) + \sin (x)$

Step 3.4.4.5

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

$\frac{4}{\cos^{2} (x)} {(\frac{\sin (x)}{\cos (x)})}^{2} + 2 \sec^{4} (x) + \sin (x)$

Step 3.4.4.6

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

$\frac{4}{\cos^{2} (x)} \cdot \frac{\sin^{2} (x)}{\cos^{2} (x)} + 2 \sec^{4} (x) + \sin (x)$

Step 3.4.4.7

Combine.

$\frac{4 \sin^{2} (x)}{\cos^{2} (x) \cos^{2} (x)} + 2 \sec^{4} (x) + \sin (x)$

Step 3.4.4.8

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

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Step 3.4.4.8.1

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

$\frac{4 \sin^{2} (x)}{{\cos (x)}^{2 + 2}} + 2 \sec^{4} (x) + \sin (x)$

Step 3.4.4.8.2

Add $2$ and $2$ .

$\frac{4 \sin^{2} (x)}{\cos^{4} (x)} + 2 \sec^{4} (x) + \sin (x)$

Step 3.4.4.9

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

$\frac{4 \sin^{2} (x)}{\cos^{4} (x)} + 2 {(\frac{1}{\cos (x)})}^{4} + \sin (x)$

Step 3.4.4.10

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

$\frac{4 \sin^{2} (x)}{\cos^{4} (x)} + 2 \frac{1^{4}}{\cos^{4} (x)} + \sin (x)$

Step 3.4.4.11

One to any power is one.

$\frac{4 \sin^{2} (x)}{\cos^{4} (x)} + 2 \frac{1}{\cos^{4} (x)} + \sin (x)$

Step 3.4.4.12

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

$\frac{4 \sin^{2} (x)}{\cos^{4} (x)} + \frac{2}{\cos^{4} (x)} + \sin (x)$

Step 3.4.5

Simplify each term.

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Step 3.4.5.1

Multiply by $1$ .

$\frac{4 (\sin^{2} (x) \cdot 1)}{\cos^{4} (x)} + \frac{2}{\cos^{4} (x)} + \sin (x)$

Step 3.4.5.2

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

$\frac{4 (\sin^{2} (x) \cdot 1)}{\cos^{2} (x) \cos^{2} (x)} + \frac{2}{\cos^{4} (x)} + \sin (x)$

Step 3.4.5.3

Separate fractions.

$\frac{4 \cdot (1)}{\cos^{2} (x)} \cdot \frac{\sin^{2} (x)}{\cos^{2} (x)} + \frac{2}{\cos^{4} (x)} + \sin (x)$

Step 3.4.5.4

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

$\frac{4 \cdot (1)}{\cos^{2} (x)} \tan^{2} (x) + \frac{2}{\cos^{4} (x)} + \sin (x)$

Step 3.4.5.5

Multiply $4$ by $1$ .

$\frac{4}{\cos^{2} (x)} \tan^{2} (x) + \frac{2}{\cos^{4} (x)} + \sin (x)$

Step 3.4.5.6

Multiply by $1$ .

$\frac{4}{\cos^{2} (x) \cdot 1} \tan^{2} (x) + \frac{2}{\cos^{4} (x)} + \sin (x)$

Step 3.4.5.7

Separate fractions.

$\frac{4}{1} \cdot \frac{1}{\cos^{2} (x)} \tan^{2} (x) + \frac{2}{\cos^{4} (x)} + \sin (x)$

Step 3.4.5.8

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

$\frac{4}{1} \sec^{2} (x) \tan^{2} (x) + \frac{2}{\cos^{4} (x)} + \sin (x)$

Step 3.4.5.9

Divide $4$ by $1$ .

$4 \sec^{2} (x) \tan^{2} (x) + \frac{2}{\cos^{4} (x)} + \sin (x)$

Step 3.4.5.10

Multiply by $1$ .

$4 \sec^{2} (x) \tan^{2} (x) + \frac{2}{\cos^{4} (x) \cdot 1} + \sin (x)$

Step 3.4.5.11

Separate fractions.

$4 \sec^{2} (x) \tan^{2} (x) + \frac{2}{1} \cdot \frac{1}{\cos^{4} (x)} + \sin (x)$

Step 3.4.5.12

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

$4 \sec^{2} (x) \tan^{2} (x) + \frac{2}{1} \sec^{4} (x) + \sin (x)$

Step 3.4.5.13

Divide $2$ by $1$ .

$f''' (x) = 4 \sec^{2} (x) \tan^{2} (x) + 2 \sec^{4} (x) + \sin (x)$

Step 4

Find the fourth derivative.

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Step 4.1

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

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

Step 4.2

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

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Step 4.2.1

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

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

Step 4.2.2

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

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 $f (x) = x^{2}$ and $g (x) = \tan (x)$ .

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Step 4.2.3.1

To apply the Chain Rule, set $u_{1}$ as $\tan (x)$ .

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

Step 4.2.3.2

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

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

Step 4.2.3.3

Replace all occurrences of $u_{1}$ with $\tan (x)$ .

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

Step 4.2.4

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

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

Step 4.2.5

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 $g (x) = \sec (x)$ .

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Step 4.2.5.1

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

$4 (\sec^{2} (x) (2 \tan (x) \sec^{2} (x)) + \tan^{2} (x) (\frac{d}{d u_{2}} [{u_{2}}^{2}] \frac{d}{d x} [\sec (x)])) + \frac{d}{d x} [2 \sec^{4} (x)] + \frac{d}{d x} [\sin (x)]$

Step 4.2.5.2

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

$4 (\sec^{2} (x) (2 \tan (x) \sec^{2} (x)) + \tan^{2} (x) (2 u_{2} \frac{d}{d x} [\sec (x)])) + \frac{d}{d x} [2 \sec^{4} (x)] + \frac{d}{d x} [\sin (x)]$

Step 4.2.5.3

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

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

Step 4.2.6

The derivative of $\sec (x)$ with respect to $x$ is $\sec (x) \tan (x)$ .

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

Step 4.2.7

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

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Step 4.2.7.1

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

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

Step 4.2.7.2

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

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

Step 4.2.7.3

Add $2$ and $2$ .

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

Step 4.2.8

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

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

Step 4.2.9

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

$4 (2 \sec^{4} (x) \tan (x) + \tan^{2} (x) (2 (\sec^{1} (x) \sec (x)) \tan (x))) + \frac{d}{d x} [2 \sec^{4} (x)] + \frac{d}{d x} [\sin (x)]$

Step 4.2.10

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

$4 (2 \sec^{4} (x) \tan (x) + \tan^{2} (x) (2 (\sec^{1} (x) \sec^{1} (x)) \tan (x))) + \frac{d}{d x} [2 \sec^{4} (x)] + \frac{d}{d x} [\sin (x)]$

Step 4.2.11

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

$4 (2 \sec^{4} (x) \tan (x) + \tan^{2} (x) (2 {\sec (x)}^{1 + 1} \tan (x))) + \frac{d}{d x} [2 \sec^{4} (x)] + \frac{d}{d x} [\sin (x)]$

Step 4.2.12

Add $1$ and $1$ .

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

Step 4.2.13

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

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Step 4.2.13.1

Move $\tan (x)$ .

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

Step 4.2.13.2

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

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Step 4.2.13.2.1

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

$4 (2 \sec^{4} (x) \tan (x) + \tan^{1} (x) \tan^{2} (x) (2 \sec^{2} (x))) + \frac{d}{d x} [2 \sec^{4} (x)] + \frac{d}{d x} [\sin (x)]$

Step 4.2.13.2.2

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

$4 (2 \sec^{4} (x) \tan (x) + {\tan (x)}^{1 + 2} (2 \sec^{2} (x))) + \frac{d}{d x} [2 \sec^{4} (x)] + \frac{d}{d x} [\sin (x)]$

Step 4.2.13.3

Add $1$ and $2$ .

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

Step 4.2.14

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

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

Step 4.3

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

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Step 4.3.1

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

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

Step 4.3.2

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

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Step 4.3.2.1

To apply the Chain Rule, set $u_{3}$ as $\sec (x)$ .

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

Step 4.3.2.2

Differentiate using the Power Rule which states that $\frac{d}{d u_{3}} [{u_{3}}^{n}]$ is $n {u_{3}}^{n - 1}$ where $n = 4$ .

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

Step 4.3.2.3

Replace all occurrences of $u_{3}$ with $\sec (x)$ .

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

Step 4.3.3

The derivative of $\sec (x)$ with respect to $x$ is $\sec (x) \tan (x)$ .

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

Step 4.3.4

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

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

Step 4.3.5

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

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

Step 4.3.6

Add $1$ and $3$ .

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

Step 4.3.7

Multiply $4$ by $2$ .

$4 (2 \sec^{4} (x) \tan (x) + 2 \tan^{3} (x) \sec^{2} (x)) + 8 \sec^{4} (x) \tan (x) + \frac{d}{d x} [\sin (x)]$

Step 4.4

The derivative of $\sin (x)$ with respect to $x$ is $\cos (x)$ .

$4 (2 \sec^{4} (x) \tan (x) + 2 \tan^{3} (x) \sec^{2} (x)) + 8 \sec^{4} (x) \tan (x) + \cos (x)$

Step 4.5

Simplify.

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Step 4.5.1

Apply the distributive property.

$4 (2 \sec^{4} (x) \tan (x)) + 4 (2 \tan^{3} (x) \sec^{2} (x)) + 8 \sec^{4} (x) \tan (x) + \cos (x)$

Step 4.5.2

Combine terms.

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Step 4.5.2.1

Multiply $2$ by $4$ .

$8 (\sec^{4} (x) \tan (x)) + 4 (2 \tan^{3} (x) \sec^{2} (x)) + 8 \sec^{4} (x) \tan (x) + \cos (x)$

Step 4.5.2.2

Multiply $2$ by $4$ .

$8 \sec^{4} (x) \tan (x) + 8 (\tan^{3} (x) \sec^{2} (x)) + 8 \sec^{4} (x) \tan (x) + \cos (x)$

Step 4.5.2.3

Add $8 \sec^{4} (x) \tan (x)$ and $8 \sec^{4} (x) \tan (x)$ .

$16 \sec^{4} (x) \tan (x) + 8 \tan^{3} (x) \sec^{2} (x) + \cos (x)$

Step 4.5.3

Simplify each term.

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Step 4.5.3.1

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

$16 {(\frac{1}{\cos (x)})}^{4} \tan (x) + 8 \tan^{3} (x) \sec^{2} (x) + \cos (x)$

Step 4.5.3.2

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

$16 \frac{1^{4}}{\cos^{4} (x)} \tan (x) + 8 \tan^{3} (x) \sec^{2} (x) + \cos (x)$

Step 4.5.3.3

One to any power is one.

$16 \frac{1}{\cos^{4} (x)} \tan (x) + 8 \tan^{3} (x) \sec^{2} (x) + \cos (x)$

Step 4.5.3.4

Combine $16$ and $\frac{1}{\cos^{4} (x)}$ .

$\frac{16}{\cos^{4} (x)} \tan (x) + 8 \tan^{3} (x) \sec^{2} (x) + \cos (x)$

Step 4.5.3.5

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

$\frac{16}{\cos^{4} (x)} \cdot \frac{\sin (x)}{\cos (x)} + 8 \tan^{3} (x) \sec^{2} (x) + \cos (x)$

Step 4.5.3.6

Combine.

$\frac{16 \sin (x)}{\cos^{4} (x) \cos (x)} + 8 \tan^{3} (x) \sec^{2} (x) + \cos (x)$

Step 4.5.3.7

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

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Step 4.5.3.7.1

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

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Step 4.5.3.7.1.1

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

$\frac{16 \sin (x)}{\cos^{4} (x) \cos^{1} (x)} + 8 \tan^{3} (x) \sec^{2} (x) + \cos (x)$

Step 4.5.3.7.1.2

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

$\frac{16 \sin (x)}{{\cos (x)}^{4 + 1}} + 8 \tan^{3} (x) \sec^{2} (x) + \cos (x)$

Step 4.5.3.7.2

Add $4$ and $1$ .

$\frac{16 \sin (x)}{\cos^{5} (x)} + 8 \tan^{3} (x) \sec^{2} (x) + \cos (x)$

Step 4.5.3.8

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

$\frac{16 \sin (x)}{\cos^{5} (x)} + 8 {(\frac{\sin (x)}{\cos (x)})}^{3} \sec^{2} (x) + \cos (x)$

Step 4.5.3.9

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

$\frac{16 \sin (x)}{\cos^{5} (x)} + 8 \frac{\sin^{3} (x)}{\cos^{3} (x)} \sec^{2} (x) + \cos (x)$

Step 4.5.3.10

Combine $8$ and $\frac{\sin^{3} (x)}{\cos^{3} (x)}$ .

$\frac{16 \sin (x)}{\cos^{5} (x)} + \frac{8 \sin^{3} (x)}{\cos^{3} (x)} \sec^{2} (x) + \cos (x)$

Step 4.5.3.11

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

$\frac{16 \sin (x)}{\cos^{5} (x)} + \frac{8 \sin^{3} (x)}{\cos^{3} (x)} {(\frac{1}{\cos (x)})}^{2} + \cos (x)$

Step 4.5.3.12

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

$\frac{16 \sin (x)}{\cos^{5} (x)} + \frac{8 \sin^{3} (x)}{\cos^{3} (x)} \cdot \frac{1^{2}}{\cos^{2} (x)} + \cos (x)$

Step 4.5.3.13

Combine.

$\frac{16 \sin (x)}{\cos^{5} (x)} + \frac{8 \sin^{3} (x) \cdot 1^{2}}{\cos^{3} (x) \cos^{2} (x)} + \cos (x)$

Step 4.5.3.14

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

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Step 4.5.3.14.1

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

$\frac{16 \sin (x)}{\cos^{5} (x)} + \frac{8 \sin^{3} (x) \cdot 1^{2}}{{\cos (x)}^{3 + 2}} + \cos (x)$

Step 4.5.3.14.2

Add $3$ and $2$ .

$\frac{16 \sin (x)}{\cos^{5} (x)} + \frac{8 \sin^{3} (x) \cdot 1^{2}}{\cos^{5} (x)} + \cos (x)$

Step 4.5.3.15

Simplify the numerator.

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Step 4.5.3.15.1

One to any power is one.

$\frac{16 \sin (x)}{\cos^{5} (x)} + \frac{8 \sin^{3} (x) \cdot 1}{\cos^{5} (x)} + \cos (x)$

Step 4.5.3.15.2

Multiply $8$ by $1$ .

$\frac{16 \sin (x)}{\cos^{5} (x)} + \frac{8 \sin^{3} (x)}{\cos^{5} (x)} + \cos (x)$

Step 4.5.4

Simplify each term.

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Step 4.5.4.1

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

$\frac{16 \sin (x)}{\cos (x) \cos^{4} (x)} + \frac{8 \sin^{3} (x)}{\cos^{5} (x)} + \cos (x)$

Step 4.5.4.2

Separate fractions.

$\frac{16}{\cos^{4} (x)} \cdot \frac{\sin (x)}{\cos (x)} + \frac{8 \sin^{3} (x)}{\cos^{5} (x)} + \cos (x)$

Step 4.5.4.3

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

$\frac{16}{\cos^{4} (x)} \tan (x) + \frac{8 \sin^{3} (x)}{\cos^{5} (x)} + \cos (x)$

Step 4.5.4.4

Multiply by $1$ .

$\frac{16}{\cos^{4} (x) \cdot 1} \tan (x) + \frac{8 \sin^{3} (x)}{\cos^{5} (x)} + \cos (x)$

Step 4.5.4.5

Separate fractions.

$\frac{16}{1} \cdot \frac{1}{\cos^{4} (x)} \tan (x) + \frac{8 \sin^{3} (x)}{\cos^{5} (x)} + \cos (x)$

Step 4.5.4.6

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

$\frac{16}{1} \sec^{4} (x) \tan (x) + \frac{8 \sin^{3} (x)}{\cos^{5} (x)} + \cos (x)$

Step 4.5.4.7

Divide $16$ by $1$ .

$16 \sec^{4} (x) \tan (x) + \frac{8 \sin^{3} (x)}{\cos^{5} (x)} + \cos (x)$

Step 4.5.4.8

Multiply by $1$ .

$16 \sec^{4} (x) \tan (x) + \frac{8 (\sin^{3} (x) \cdot 1)}{\cos^{5} (x)} + \cos (x)$

Step 4.5.4.9

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

$16 \sec^{4} (x) \tan (x) + \frac{8 (\sin^{3} (x) \cdot 1)}{\cos^{3} (x) \cos^{2} (x)} + \cos (x)$

Step 4.5.4.10

Separate fractions.

$16 \sec^{4} (x) \tan (x) + \frac{8 \cdot (1)}{\cos^{2} (x)} \cdot \frac{\sin^{3} (x)}{\cos^{3} (x)} + \cos (x)$

Step 4.5.4.11

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

$16 \sec^{4} (x) \tan (x) + \frac{8 \cdot (1)}{\cos^{2} (x)} \tan^{3} (x) + \cos (x)$

Step 4.5.4.12

Multiply $8$ by $1$ .

$16 \sec^{4} (x) \tan (x) + \frac{8}{\cos^{2} (x)} \tan^{3} (x) + \cos (x)$

Step 4.5.4.13

Multiply by $1$ .

$16 \sec^{4} (x) \tan (x) + \frac{8}{\cos^{2} (x) \cdot 1} \tan^{3} (x) + \cos (x)$

Step 4.5.4.14

Separate fractions.

$16 \sec^{4} (x) \tan (x) + \frac{8}{1} \cdot \frac{1}{\cos^{2} (x)} \tan^{3} (x) + \cos (x)$

Step 4.5.4.15

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

$16 \sec^{4} (x) \tan (x) + \frac{8}{1} \sec^{2} (x) \tan^{3} (x) + \cos (x)$

Step 4.5.4.16

Divide $8$ by $1$ .

$f^{4} (x) = 16 \sec^{4} (x) \tan (x) + 8 \sec^{2} (x) \tan^{3} (x) + \cos (x)$