Calculus Examples

$f (x) = x - 2 \sin (x)$

Step 1

Find the first derivative.

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Step 1.1

Differentiate.

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Step 1.1.1

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

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

Step 1.1.2

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

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

Step 1.2

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

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Step 1.2.1

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

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

Step 1.2.2

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

$f' (x) = 1 - 2 \cos (x)$

Step 2

Find the second derivative.

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Step 2.1

Differentiate.

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Step 2.1.1

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

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

Step 2.1.2

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

$0 + \frac{d}{d x} [- 2 \cos (x)]$

Step 2.2

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

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Step 2.2.1

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

$0 - 2 \frac{d}{d x} [\cos (x)]$

Step 2.2.2

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

$0 - 2 (- \sin (x))$

Step 2.2.3

Multiply $- 1$ by $- 2$ .

$0 + 2 \sin (x)$

Step 2.3

Add $0$ and $2 \sin (x)$ .

$f'' (x) = 2 \sin (x)$

Step 3

Find the third derivative.

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Step 3.1

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

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

Step 3.2

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

$f''' (x) = 2 \cos (x)$

Step 4

Find the fourth derivative.

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Step 4.1

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

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

Step 4.2

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

$2 (- \sin (x))$

Step 4.3

Multiply $- 1$ by $2$ .

$f^{4} (x) = - 2 \sin (x)$

Step 5

The fourth derivative of $f (x)$ with respect to $x$ is $- 2 \sin (x)$ .

$- 2 \sin (x)$