Calculus Examples

$2 \cos^{2} (x^{2} + 2)$

Step 1

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

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

Step 2

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

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Step 2.1

To apply the Chain Rule, set $u_{1}$ as $\cos (x^{2} + 2)$ .

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

Step 2.2

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

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

Step 2.3

Replace all occurrences of $u_{1}$ with $\cos (x^{2} + 2)$ .

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

Step 3

Multiply $2$ by $2$ .

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

Step 4

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 $g (x) = x^{2} + 2$ .

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Step 4.1

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

$4 \cos (x^{2} + 2) (\frac{d}{d u_{2}} [\cos (u_{2})] \frac{d}{d x} [x^{2} + 2])$

Step 4.2

The derivative of $\cos (u_{2})$ with respect to $u_{2}$ is $- \sin (u_{2})$ .

$4 \cos (x^{2} + 2) (- \sin (u_{2}) \frac{d}{d x} [x^{2} + 2])$

Step 4.3

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

$4 \cos (x^{2} + 2) (- \sin (x^{2} + 2) \frac{d}{d x} [x^{2} + 2])$

Step 5

Differentiate.

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Step 5.1

Multiply $- 1$ by $4$ .

$- 4 \cos (x^{2} + 2) (\sin (x^{2} + 2) \frac{d}{d x} [x^{2} + 2])$

Step 5.2

By the Sum Rule, the derivative of $x^{2} + 2$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [2]$ .

$- 4 \cos (x^{2} + 2) \sin (x^{2} + 2) (\frac{d}{d x} [x^{2}] + \frac{d}{d x} [2])$

Step 5.3

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

$- 4 \cos (x^{2} + 2) \sin (x^{2} + 2) (2 x + \frac{d}{d x} [2])$

Step 5.4

Since $2$ is constant with respect to $x$ , the derivative of $2$ with respect to $x$ is $0$ .

$- 4 \cos (x^{2} + 2) \sin (x^{2} + 2) (2 x + 0)$

Step 5.5

Simplify the expression.

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Step 5.5.1

Add $2 x$ and $0$ .

$- 4 \cos (x^{2} + 2) \sin (x^{2} + 2) (2 x)$

Step 5.5.2

Multiply $2$ by $- 4$ .

$- 8 \cos (x^{2} + 2) \sin (x^{2} + 2) x$

Step 5.5.3

Reorder the factors of $- 8 \cos (x^{2} + 2) \sin (x^{2} + 2) x$ .

$- 8 x \cos (x^{2} + 2) \sin (x^{2} + 2)$