Calculus Examples

$\sin (\frac{π}{2} x)$

Step 1

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

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Step 1.1

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

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

Step 1.2

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

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

Step 1.3

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

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

Step 2

Differentiate.

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Step 2.1

Combine $\frac{π}{2}$ and $x$ .

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

Step 2.2

Combine $\frac{π}{2}$ and $x$ .

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

Step 2.3

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

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

Step 2.4

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

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

Step 2.5

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

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

Step 2.6

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

$\frac{π \cos (\frac{π x}{2})}{2}$