Calculus Examples

$f (x) = 4 \tan (x^{2})$

Step 1

Since $4$ is constant with respect to $x$ , the derivative of $4 \tan (x^{2})$ with respect to $x$ is $4 \frac{d}{d x} [\tan (x^{2})]$ .

$4 \frac{d}{d x} [\tan (x^{2})]$

Step 2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \tan (x)$ and $g (x) = x^{2}$ .

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Step 2.1

To apply the Chain Rule, set $u$ as $x^{2}$ .

$4 (\frac{d}{d u} [\tan (u)] \frac{d}{d x} [x^{2}])$

Step 2.2

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

$4 (\sec^{2} (u) \frac{d}{d x} [x^{2}])$

Step 2.3

Replace all occurrences of $u$ with $x^{2}$ .

$4 (\sec^{2} (x^{2}) \frac{d}{d x} [x^{2}])$

Step 3

Differentiate using the Power Rule.

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Step 3.1

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

$4 \sec^{2} (x^{2}) (2 x)$

Step 3.2

Simplify the expression.

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Step 3.2.1

Multiply $2$ by $4$ .

$8 \sec^{2} (x^{2}) x$

Step 3.2.2

Reorder the factors of $8 \sec^{2} (x^{2}) x$ .

$8 x \sec^{2} (x^{2})$