Calculus Examples

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

Step 1

By the Sum Rule, the derivative of $\frac{1}{2} x - \sin (x)$ with respect to $x$ is $\frac{d}{d x} [\frac{1}{2} x] + \frac{d}{d x} [- \sin (x)]$ .

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

Step 2

Evaluate $\frac{d}{d x} [\frac{1}{2} x]$ .

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

Multiply $\frac{1}{2}$ by $1$ .

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

Step 3

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

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Step 3.1

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

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

Step 3.2

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

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