Calculus Examples

$\cos (2 \arctan (x))$

Step 1

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

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Step 1.1

To apply the Chain Rule, set $u$ as $2 \arctan (x)$ .

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

Step 1.2

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

$- \sin (u) \frac{d}{d x} [2 \arctan (x)]$

Step 1.3

Replace all occurrences of $u$ with $2 \arctan (x)$ .

$- \sin (2 \arctan (x)) \frac{d}{d x} [2 \arctan (x)]$

Step 2

Differentiate using the Constant Multiple Rule.

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Step 2.1

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

$- \sin (2 \arctan (x)) (2 \frac{d}{d x} [\arctan (x)])$

Step 2.2

Multiply $2$ by $- 1$ .

$- 2 \sin (2 \arctan (x)) \frac{d}{d x} [\arctan (x)]$

Step 3

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

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

Step 4

Combine fractions.

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Step 4.1

Combine $\frac{1}{1 + x^{2}}$ and $- 2$ .

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

Step 4.2

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

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

Step 4.3

Move the negative in front of the fraction.

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