Calculus Examples

$2 \sin (x)$

Step 1

Find the first derivative.

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Step 1.1

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 1.2

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

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

Step 2

Find the second derivative.

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Step 2.1

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

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

Step 2.2

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

$2 (- \sin (x))$

Step 2.3

Multiply $- 1$ by $2$ .

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

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 $- 2$ .

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