Calculus Examples

$x - \sin (x)$

Step 1

Find the first derivative.

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Differentiate.

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

$f' (x) = \frac{d}{d x} (x) + \frac{d}{d x} (- \sin (x))$

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

$f' (x) = 1 + \frac{d}{d x} (- \sin (x))$

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

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

$f' (x) = 1 - \frac{d}{d x} \sin (x)$

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

$f' (x) = 1 - \cos (x)$

Step 2

Find the second derivative.

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Differentiate.

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

$f'' (x) = \frac{d}{d x} (1) + \frac{d}{d x} (- \cos (x))$

Since $1$ is constant with respect to $x$ , the derivative of $1$ with respect to $x$ is $0$ .

$f'' (x) = 0 + \frac{d}{d x} (- \cos (x))$

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

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

$f'' (x) = 0 - \frac{d}{d x} \cos (x)$

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

$f'' (x) = 0 + \sin (x)$

Multiply $- 1$ by $- 1$ .

$f'' (x) = 0 + 1 \sin (x)$

Multiply $\sin (x)$ by $1$ .

$f'' (x) = 0 + \sin (x)$

Add $0$ and $\sin (x)$ .

$f'' (x) = \sin (x)$

Step 3

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

$f''' (x) = \cos (x)$

Step 4

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

$f^{4} (x) = - \sin (x)$