Calculus Examples

$f (x) = (\cos (x)) \cdot 3$

Step 1

Find the first derivative.

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Step 1.1

Differentiate using the Constant Multiple Rule.

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Step 1.1.1

Move $3$ to the left of $\cos (x)$ .

$\frac{d}{d x} [3 \cdot \cos (x)]$

Step 1.1.2

Since $3$ is constant with respect to $x$ , the derivative of $3 \cos (x)$ with respect to $x$ is $3 \frac{d}{d x} [\cos (x)]$ .

$3 \frac{d}{d x} [\cos (x)]$

Step 1.2

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

$3 (- \sin (x))$

Step 1.3

Multiply $- 1$ by $3$ .

$f' (x) = - 3 \sin (x)$

Step 2

Find the second derivative.

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Step 2.1

Since $- 3$ is constant with respect to $x$ , the derivative of $- 3 \sin (x)$ with respect to $x$ is $- 3 \frac{d}{d x} [\sin (x)]$ .

$- 3 \frac{d}{d x} [\sin (x)]$

Step 2.2

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

$f'' (x) = - 3 \cos (x)$

Step 3

Find the third derivative.

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Step 3.1

Since $- 3$ is constant with respect to $x$ , the derivative of $- 3 \cos (x)$ with respect to $x$ is $- 3 \frac{d}{d x} [\cos (x)]$ .

$- 3 \frac{d}{d x} [\cos (x)]$

Step 3.2

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

$- 3 (- \sin (x))$

Step 3.3

Multiply $- 1$ by $- 3$ .

$f''' (x) = 3 \sin (x)$

Step 4

The third derivative of $f (x)$ with respect to $x$ is $3 \sin (x)$ .

$3 \sin (x)$