Calculus Examples

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

Step 1

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

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Step 1.1

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

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

Step 1.2

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

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

Step 1.3

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

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

Step 2

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

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Step 2.1

To apply the Chain Rule, set $u_{2}$ as $\frac{x}{2}$ .

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

Step 2.2

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

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

Step 2.3

Replace all occurrences of $u_{2}$ with $\frac{x}{2}$ .

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

Step 3

Differentiate.

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Step 3.1

Multiply $- 1$ by $2$ .

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

Step 3.2

Since $\frac{1}{2}$ is constant with respect to $x$ , the derivative of $\frac{x}{2}$ with respect to $x$ is $\frac{1}{2} \frac{d}{d x} [x]$ .

$- 2 \cos (\frac{x}{2}) \sin (\frac{x}{2}) (\frac{1}{2} \frac{d}{d x} [x])$

Step 3.3

Simplify terms.

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Step 3.3.1

Combine $\frac{1}{2}$ and $- 2$ .

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

Step 3.3.2

Combine $\frac{- 2}{2}$ and $\cos (\frac{x}{2})$ .

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

Step 3.3.3

Combine $\frac{- 2 \cos (\frac{x}{2})}{2}$ and $\sin (\frac{x}{2})$ .

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

Step 3.3.4

Cancel the common factor of $- 2$ and $2$ .

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Step 3.3.4.1

Factor $2$ out of $- 2 \cos (\frac{x}{2}) \sin (\frac{x}{2})$ .

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

Step 3.3.4.2

Cancel the common factors.

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Step 3.3.4.2.1

Factor $2$ out of $2$ .

$\frac{2 (- \cos (\frac{x}{2}) \sin (\frac{x}{2}))}{2 (1)} \frac{d}{d x} [x]$

Step 3.3.4.2.2

Cancel the common factor.

$\frac{2 (- \cos (\frac{x}{2}) \sin (\frac{x}{2}))}{2 \cdot 1} \frac{d}{d x} [x]$

Step 3.3.4.2.3

Rewrite the expression.

$\frac{- \cos (\frac{x}{2}) \sin (\frac{x}{2})}{1} \frac{d}{d x} [x]$

Step 3.3.4.2.4

Divide $- \cos (\frac{x}{2}) \sin (\frac{x}{2})$ by $1$ .

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

Step 3.4

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

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

Step 3.5

Multiply $- 1$ by $1$ .

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