Calculus Examples

$y = \frac{4 \sin (2 x)}{3 \tan (3 x)} - \cos^{2} (3 x)$

Step 1

Separate fractions.

$\frac{d}{d y} [\frac{4}{3} \cdot \frac{\sin (2 x)}{\tan (3 x)} - \cos^{2} (3 x)]$

Step 2

Rewrite $\tan (3 x)$ in terms of sines and cosines.

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

Step 3

Multiply by the reciprocal of the fraction to divide by $\frac{\sin (3 x)}{\cos (3 x)}$ .

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

Step 4

Write $\sin (2 x)$ as a fraction with denominator $1$ .

$\frac{d}{d y} [\frac{4}{3} (\frac{\sin (2 x)}{1} \cdot \frac{\cos (3 x)}{\sin (3 x)}) - \cos^{2} (3 x)]$

Step 5

Simplify.

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Step 5.1

Divide $\sin (2 x)$ by $1$ .

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

Step 5.2

Combine $\sin (2 x)$ and $\frac{\cos (3 x)}{\sin (3 x)}$ .

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

Step 6

Multiply $\frac{4}{3}$ by $\frac{\sin (2 x) \cos (3 x)}{\sin (3 x)}$ .

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

Step 7

Separate fractions.

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

Step 8

Rewrite $\frac{\sin (2 x)}{\sin (3 x)}$ as a product.

$\frac{d}{d y} [\frac{4 \cos (3 x)}{3} (\sin (2 x) \frac{1}{\sin (3 x)}) - \cos^{2} (3 x)]$

Step 9

Write $\sin (2 x)$ as a fraction with denominator $1$ .

$\frac{d}{d y} [\frac{4 \cos (3 x)}{3} (\frac{\sin (2 x)}{1} \cdot \frac{1}{\sin (3 x)}) - \cos^{2} (3 x)]$

Step 10

Simplify.

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Step 10.1

Divide $\sin (2 x)$ by $1$ .

$\frac{d}{d y} [\frac{4 \cos (3 x)}{3} (\sin (2 x) \frac{1}{\sin (3 x)}) - \cos^{2} (3 x)]$

Step 10.2

Convert from $\frac{1}{\sin (3 x)}$ to $\csc (3 x)$ .

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

Step 11

Multiply $\frac{4 \cos (3 x)}{3} (\sin (2 x) \csc (3 x))$ .

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Step 11.1

Combine $\sin (2 x)$ and $\frac{4 \cos (3 x)}{3}$ .

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

Step 11.2

Combine $\frac{\sin (2 x) (4 \cos (3 x))}{3}$ and $\csc (3 x)$ .

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

Step 12

Move $4$ to the left of $\sin (2 x)$ .

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

Step 13

By the Sum Rule, the derivative of $\frac{4 \sin (2 x) \cos (3 x) \csc (3 x)}{3} - \cos^{2} (3 x)$ with respect to $y$ is $\frac{d}{d y} [\frac{4 \sin (2 x) \cos (3 x) \csc (3 x)}{3}] + \frac{d}{d y} [- \cos^{2} (3 x)]$ .

$\frac{d}{d y} [\frac{4 \sin (2 x) \cos (3 x) \csc (3 x)}{3}] + \frac{d}{d y} [- \cos^{2} (3 x)]$

Step 14

Since $\frac{4 \sin (2 x) \cos (3 x) \csc (3 x)}{3}$ is constant with respect to $y$ , the derivative of $\frac{4 \sin (2 x) \cos (3 x) \csc (3 x)}{3}$ with respect to $y$ is $0$ .

$0 + \frac{d}{d y} [- \cos^{2} (3 x)]$

Step 15

Since $- \cos^{2} (3 x)$ is constant with respect to $y$ , the derivative of $- \cos^{2} (3 x)$ with respect to $y$ is $0$ .

$0 + 0$

Step 16

Add $0$ and $0$ .

$0$