Calculus Examples

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

Step 1

Differentiate both sides of the equation.

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

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...

Step 3.1

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = \sin (6 x)$ and $g (x) = \cos^{2} (x)$ .

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

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 $f (x) = x^{2}$ and $g (x) = \cos (x)$ .

Tap for more steps...

Step 3.2.1

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

$\sin (6 x) (\frac{d}{d u_{1}} [{u_{1}}^{2}] \frac{d}{d x} [\cos (x)]) + \cos^{2} (x) \frac{d}{d x} [\sin (6 x)]$

Step 3.2.2

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

$\sin (6 x) (2 u_{1} \frac{d}{d x} [\cos (x)]) + \cos^{2} (x) \frac{d}{d x} [\sin (6 x)]$

Step 3.2.3

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

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

Step 3.3

Move $2$ to the left of $\sin (6 x)$ .

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

Step 3.4

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

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

Step 3.5

Multiply $- 1$ by $2$ .

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

Step 3.6

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

Tap for more steps...

Step 3.6.1

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

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

Step 3.6.2

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

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

Step 3.6.3

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

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

Step 3.7

Differentiate.

Tap for more steps...

Step 3.7.1

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

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

Step 3.7.2

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

$- 2 \sin (6 x) \cos (x) \sin (x) + \cos^{2} (x) \cos (6 x) (6 \cdot 1)$

Step 3.7.3

Simplify the expression.

Tap for more steps...

Step 3.7.3.1

Multiply $6$ by $1$ .

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

Step 3.7.3.2

Move $6$ to the left of $\cos^{2} (x) \cos (6 x)$ .

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

Step 3.7.3.3

Reorder terms.

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

Step 4

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

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

Step 5

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

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