Calculus Examples

$y = \frac{\sin^{2} (x)}{\cos^{2} (x)}$

Step 1

Convert from $\frac{\sin^{2} (x)}{\cos^{2} (x)}$ to $\tan^{2} (x)$ .

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

Step 2

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 2.1

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

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

Step 2.2

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 2.3

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

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

Step 3

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

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

Step 4

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

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