Calculus Examples

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

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (y) = \frac{d}{d x} (2 \cos^{3} (x))$

Step 2

The derivative of $y$ with respect to $x$ is $y'$ .

$y'$

Step 3

Differentiate the right side of the equation.

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Step 3.1

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

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

Step 3.2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{3}$ and $g (x) = \cos (x)$ .

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Step 3.2.1

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

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

Step 3.2.2

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = 3$ .

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

Step 3.2.3

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

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

Step 3.3

Multiply $3$ by $2$ .

$6 (\cos^{2} (x) \frac{d}{d x} [\cos (x)])$

Step 3.4

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

$6 \cos^{2} (x) (- \sin (x))$

Step 3.5

Multiply $- 1$ by $6$ .

$- 6 \cos^{2} (x) \sin (x)$

Step 4

Reform the equation by setting the left side equal to the right side.

$y' = - 6 \cos^{2} (x) \sin (x)$

Step 5

Replace $y'$ with $\frac{d y}{d x}$ .

$\frac{d y}{d x} = - 6 \cos^{2} (x) \sin (x)$