Calculus Examples

$y = \cos^{4} (6 x)$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (y) = \frac{d}{d x} (\cos^{4} (6 x))$

Step 2

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

$y'$

Step 3

Differentiate the right side of the equation.

Tap for more steps...

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{4}$ and $g (x) = \cos (6 x)$ .

Tap for more steps...

To apply the Chain Rule, set $u_{1}$ as $\cos (6 x)$ .

$\frac{d}{d u_{1}} [{u_{1}}^{4}] \frac{d}{d x} [\cos (6 x)]$

Differentiate using the Power Rule which states that $\frac{d}{d u_{1}} [{u_{1}}^{n}]$ is $n {u_{1}}^{n - 1}$ where $n = 4$ .

$4 {u_{1}}^{3} \frac{d}{d x} [\cos (6 x)]$

Replace all occurrences of $u_{1}$ with $\cos (6 x)$ .

$4 \cos^{3} (6 x) \frac{d}{d x} [\cos (6 x)]$

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \cos (x)$ and $g (x) = 6 x$ .

Tap for more steps...

To apply the Chain Rule, set $u_{2}$ as $6 x$ .

$4 \cos^{3} (6 x) (\frac{d}{d u_{2}} [\cos (u_{2})] \frac{d}{d x} [6 x])$

The derivative of $\cos (u_{2})$ with respect to $u_{2}$ is $- \sin (u_{2})$ .

$4 \cos^{3} (6 x) (- \sin (u_{2}) \frac{d}{d x} [6 x])$

Replace all occurrences of $u_{2}$ with $6 x$ .

$4 \cos^{3} (6 x) (- \sin (6 x) \frac{d}{d x} [6 x])$

Differentiate.

Tap for more steps...

Multiply $- 1$ by $4$ .

$- 4 \cos^{3} (6 x) (\sin (6 x) \frac{d}{d x} [6 x])$

Since $6$ is constant with respect to $x$ , the derivative of $6 x$ with respect to $x$ is $6 \frac{d}{d x} [x]$ .

$- 4 \cos^{3} (6 x) \sin (6 x) (6 \frac{d}{d x} [x])$

Multiply $6$ by $- 4$ .

$- 24 \cos^{3} (6 x) \sin (6 x) \frac{d}{d x} [x]$

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

$- 24 \cos^{3} (6 x) \sin (6 x) \cdot 1$

Multiply $- 24$ by $1$ .

$- 24 \cos^{3} (6 x) \sin (6 x)$

Step 4

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

$y' = - 24 \cos^{3} (6 x) \sin (6 x)$

Step 5

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

$\frac{d y}{d x} = - 24 \cos^{3} (6 x) \sin (6 x)$