Calculus Examples

y = sec (tan (x))

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

Step 1

Differentiate both sides of the equation.

\frac{d}{d x} (y) = \frac{d}{d x} (sec (tan (x)))

$\frac{d}{d x} (y) = \frac{d}{d x} (\sec (\tan (x)))$

Step 2

The derivative of

y

$y$ with respect to

x

$x$ is

$y'$ .

$y'$

Step 3

Differentiate the right side of the equation.

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Step 3.1

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 3.1.1

To apply the Chain Rule, set

$u$ as

$\tan (x)$ .

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

Step 3.1.2

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 3.1.3

Replace all occurrences of

$u$ with

$\tan (x)$ .

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

Step 3.2

The derivative of

$\tan (x)$ with respect to

$x$ is

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

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

Step 3.3

Reorder the factors of

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

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

Step 4

Reform the equation by setting the left side equal to the right side.

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

Step 5

Replace

$y'$ with

$\frac{d y}{d x}$ .

$\frac{d y}{d x} = \sec^{2} (x) \sec (\tan (x)) \tan (\tan (x))$

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

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