Calculus Examples

$y = \sec (\tan (x))$

Step 1

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

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Step 1.1

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

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

Step 1.2

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

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

Step 1.3

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

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

Step 2

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

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

Step 3

Reorder the factors of $\sec (\tan (x)) \tan (\tan (x)) \sec^{2} (x)$ .

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