Calculus Examples

$3 \cos^{2} (x)$

Step 1

Since $3$ is constant with respect to $x$ , the derivative of $3 \cos^{2} (x)$ with respect to $x$ is $3 \frac{d}{d x} [\cos^{2} (x)]$ .

$3 \frac{d}{d x} [\cos^{2} (x)]$

Step 2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{2}$ and $g (x) = \cos (x)$ .

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Step 2.1

To apply the Chain Rule, set $u$ as $\cos (x)$ .

$3 (\frac{d}{d u} [u^{2}] \frac{d}{d x} [\cos (x)])$

Step 2.2

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = 2$ .

$3 (2 u \frac{d}{d x} [\cos (x)])$

Step 2.3

Replace all occurrences of $u$ with $\cos (x)$ .

$3 (2 \cos (x) \frac{d}{d x} [\cos (x)])$

Step 3

Multiply $2$ by $3$ .

$6 (\cos (x) \frac{d}{d x} [\cos (x)])$

Step 4

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

$6 \cos (x) (- \sin (x))$

Step 5

Multiply $- 1$ by $6$ .

$- 6 \cos (x) \sin (x)$