Calculus Examples

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

Step 1

Differentiate both sides of the equation.

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

Step 2

The derivative of $y$ with respect to $x$ 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 $f (x) = x^{2}$ and $g (x) = \tan (x)$ .

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Step 3.1.1

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

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

Step 3.1.2

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

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

Step 3.1.3

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

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

Step 3.2

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

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

Step 3.3

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

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

Step 4

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

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

Step 5

Replace $y'$ with $\frac{d y}{d x}$ .

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