Calculus Examples

$\arctan (\cos (x))$

Step 1

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

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Step 1.1

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

$\frac{d}{d u} [\arctan (u)] \frac{d}{d x} [\cos (x)]$

Step 1.2

The derivative of $\arctan (u)$ with respect to $u$ is $\frac{1}{1 + u^{2}}$ .

$\frac{1}{1 + u^{2}} \frac{d}{d x} [\cos (x)]$

Step 1.3

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

$\frac{1}{1 + \cos^{2} (x)} \frac{d}{d x} [\cos (x)]$

Step 2

The derivative of $\cos (x)$ with respect to $x$ is $- \sin (x)$ .

$\frac{1}{1 + \cos^{2} (x)} (- \sin (x))$

Step 3

Combine $\frac{1}{1 + \cos^{2} (x)}$ and $\sin (x)$ .

$- \frac{\sin (x)}{1 + \cos^{2} (x)}$