Calculus Examples

$y = \tan (x)$

Step 1

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

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

Step 2

Find the second derivative.

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Step 2.1

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 2.1.1

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

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

Step 2.1.2

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

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

Step 2.1.3

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

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

Step 2.2

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

$2 \sec (x) (\sec (x) \tan (x))$

Step 2.3

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

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

Step 2.4

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

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

Step 2.5

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

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

Step 2.6

Add $1$ and $1$ .

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