Calculus Examples

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

Step 1

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

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

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 $u$ as $\frac{x}{2}$ .

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

Step 2.2

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

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

Step 2.3

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

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

Step 3

Differentiate.

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Step 3.1

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

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

Step 3.2

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

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

Step 3.3

Combine fractions.

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Step 3.3.1

Multiply $\frac{1}{2}$ by $\frac{\sin (\frac{x}{2})}{2}$ .

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

Step 3.3.2

Multiply $2$ by $2$ .

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

Step 3.4

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

$- \frac{\sin (\frac{x}{2})}{4} \cdot 1$

Step 3.5

Multiply $- 1$ by $1$ .

$- \frac{\sin (\frac{x}{2})}{4}$