Calculus Examples

$y = 2 \sec (5 x)$

Step 1

Find the first derivative.

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Step 1.1

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

$2 \frac{d}{d x} [\sec (5 x)]$

Step 1.2

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

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Step 1.2.1

To apply the Chain Rule, set $u$ as $5 x$ .

$2 (\frac{d}{d u} [\sec (u)] \frac{d}{d x} [5 x])$

Step 1.2.2

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

$2 (\sec (u) \tan (u) \frac{d}{d x} [5 x])$

Step 1.2.3

Replace all occurrences of $u$ with $5 x$ .

$2 (\sec (5 x) \tan (5 x) \frac{d}{d x} [5 x])$

Step 1.3

Differentiate.

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Step 1.3.1

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

$2 \sec (5 x) \tan (5 x) (5 \frac{d}{d x} [x])$

Step 1.3.2

Multiply $5$ by $2$ .

$10 \sec (5 x) \tan (5 x) \frac{d}{d x} [x]$

Step 1.3.3

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

$10 \sec (5 x) \tan (5 x) \cdot 1$

Step 1.3.4

Multiply $10$ by $1$ .

$f' (x) = 10 \sec (5 x) \tan (5 x)$

Step 2

Find the second derivative.

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Step 2.1

Since $10$ is constant with respect to $x$ , the derivative of $10 \sec (5 x) \tan (5 x)$ with respect to $x$ is $10 \frac{d}{d x} [\sec (5 x) \tan (5 x)]$ .

$10 \frac{d}{d x} [\sec (5 x) \tan (5 x)]$

Step 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 $f (x) = \sec (5 x)$ and $g (x) = \tan (5 x)$ .

$10 (\sec (5 x) \frac{d}{d x} [\tan (5 x)] + \tan (5 x) \frac{d}{d x} [\sec (5 x)])$

Step 2.3

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

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Step 2.3.1

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

$10 (\sec (5 x) (\frac{d}{d u_{1}} [\tan (u_{1})] \frac{d}{d x} [5 x]) + \tan (5 x) \frac{d}{d x} [\sec (5 x)])$

Step 2.3.2

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

$10 (\sec (5 x) (\sec^{2} (u_{1}) \frac{d}{d x} [5 x]) + \tan (5 x) \frac{d}{d x} [\sec (5 x)])$

Step 2.3.3

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

$10 (\sec (5 x) (\sec^{2} (5 x) \frac{d}{d x} [5 x]) + \tan (5 x) \frac{d}{d x} [\sec (5 x)])$

Step 2.4

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

$10 (\sec^{2} (5 x) \sec^{1} (5 x) \frac{d}{d x} [5 x] + \tan (5 x) \frac{d}{d x} [\sec (5 x)])$

Step 2.5

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

$10 ({\sec (5 x)}^{2 + 1} \frac{d}{d x} [5 x] + \tan (5 x) \frac{d}{d x} [\sec (5 x)])$

Step 2.6

Differentiate.

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Step 2.6.1

Add $2$ and $1$ .

$10 (\sec^{3} (5 x) \frac{d}{d x} [5 x] + \tan (5 x) \frac{d}{d x} [\sec (5 x)])$

Step 2.6.2

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

$10 (\sec^{3} (5 x) (5 \frac{d}{d x} [x]) + \tan (5 x) \frac{d}{d x} [\sec (5 x)])$

Step 2.6.3

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

$10 (\sec^{3} (5 x) (5 \cdot 1) + \tan (5 x) \frac{d}{d x} [\sec (5 x)])$

Step 2.6.4

Simplify the expression.

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Step 2.6.4.1

Multiply $5$ by $1$ .

$10 (\sec^{3} (5 x) \cdot 5 + \tan (5 x) \frac{d}{d x} [\sec (5 x)])$

Step 2.6.4.2

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

$10 (5 \cdot \sec^{3} (5 x) + \tan (5 x) \frac{d}{d x} [\sec (5 x)])$

Step 2.7

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

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Step 2.7.1

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

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

Step 2.7.2

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

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

Step 2.7.3

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

$10 (5 \sec^{3} (5 x) + \tan (5 x) (\sec (5 x) \tan (5 x) \frac{d}{d x} [5 x]))$

Step 2.8

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

$10 (5 \sec^{3} (5 x) + \tan^{1} (5 x) \tan (5 x) (\sec (5 x) \frac{d}{d x} [5 x]))$

Step 2.9

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

$10 (5 \sec^{3} (5 x) + \tan^{1} (5 x) \tan^{1} (5 x) (\sec (5 x) \frac{d}{d x} [5 x]))$

Step 2.10

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

$10 (5 \sec^{3} (5 x) + {\tan (5 x)}^{1 + 1} (\sec (5 x) \frac{d}{d x} [5 x]))$

Step 2.11

Add $1$ and $1$ .

$10 (5 \sec^{3} (5 x) + \tan^{2} (5 x) (\sec (5 x) \frac{d}{d x} [5 x]))$

Step 2.12

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

$10 (5 \sec^{3} (5 x) + \tan^{2} (5 x) \sec (5 x) (5 \frac{d}{d x} [x]))$

Step 2.13

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

$10 (5 \sec^{3} (5 x) + \tan^{2} (5 x) \sec (5 x) (5 \cdot 1))$

Step 2.14

Simplify the expression.

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Step 2.14.1

Multiply $5$ by $1$ .

$10 (5 \sec^{3} (5 x) + \tan^{2} (5 x) \sec (5 x) \cdot 5)$

Step 2.14.2

Move $5$ to the left of $\tan^{2} (5 x) \sec (5 x)$ .

$10 (5 \sec^{3} (5 x) + 5 \cdot (\tan^{2} (5 x) \sec (5 x)))$

Step 2.15

Simplify.

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Step 2.15.1

Apply the distributive property.

$10 (5 \sec^{3} (5 x)) + 10 (5 \tan^{2} (5 x) \sec (5 x))$

Step 2.15.2

Combine terms.

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Step 2.15.2.1

Multiply $5$ by $10$ .

$50 \sec^{3} (5 x) + 10 (5 \tan^{2} (5 x) \sec (5 x))$

Step 2.15.2.2

Multiply $5$ by $10$ .

$50 \sec^{3} (5 x) + 50 \tan^{2} (5 x) \sec (5 x)$

Step 2.15.3

Reorder terms.

$f'' (x) = 50 \tan^{2} (5 x) \sec (5 x) + 50 \sec^{3} (5 x)$