Calculus Examples

$f (x) = 2 \sec (x)$

Find the first derivative.

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

$f' (x) = 2 \frac{d}{d x} (\sec (x))$

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

$f' (x) = 2 \sec (x) \tan (x)$

Find the second derivative.

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

$f'' (x) = 2 \frac{d}{d x} (\sec (x) \tan (x))$

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

$f'' (x) = 2 (\sec (x) \frac{d}{d x} (\tan (x)) + \tan (x) \frac{d}{d x} (\sec (x)))$

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

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

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

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

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

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

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

$f'' (x) = 2 ({\sec (x)}^{1 + 2} + \tan (x) \frac{d}{d x} (\sec (x)))$

Add $1$ and $2$ .

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

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

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

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

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

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

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

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

$f'' (x) = 2 (\sec^{3} (x) + {\tan (x)}^{1 + 1} \sec (x))$

Add $1$ and $1$ .

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

Simplify.

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Apply the distributive property.

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

Reorder terms.

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

The second derivative of $f (x)$ with respect to $x$ is $2 \tan^{2} (x) \sec (x) + 2 \sec^{3} (x)$ .

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