Calculus Examples

$y = 3 \sec (x)$

Step 1

Find the first derivative.

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Step 1.1

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

$3 \frac{d}{d x} [\sec (x)]$

Step 1.2

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

$f' (x) = 3 \sec (x) \tan (x)$

Step 2

Find the second derivative.

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Step 2.1

Since $3$ is constant with respect to $x$ , the derivative of $3 \sec (x) \tan (x)$ with respect to $x$ is $3 \frac{d}{d x} [\sec (x) \tan (x)]$ .

$3 \frac{d}{d x} [\sec (x) \tan (x)]$

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

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

Step 2.3

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

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

Step 2.4

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

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Step 2.4.1

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

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Step 2.4.1.1

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

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

Step 2.4.1.2

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

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

Step 2.4.2

Add $1$ and $2$ .

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

Step 2.5

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

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

Step 2.6

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

$3 (\sec^{3} (x) + \tan^{1} (x) \tan (x) \sec (x))$

Step 2.7

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

$3 (\sec^{3} (x) + \tan^{1} (x) \tan^{1} (x) \sec (x))$

Step 2.8

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

$3 (\sec^{3} (x) + {\tan (x)}^{1 + 1} \sec (x))$

Step 2.9

Add $1$ and $1$ .

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

Step 2.10

Simplify.

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Step 2.10.1

Apply the distributive property.

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

Step 2.10.2

Reorder terms.

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