Calculus Examples

$y = \frac{1}{\sin (20 x)}$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d y} (y) = \frac{d}{d y} (\frac{1}{\sin (20 x)})$

Step 2

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

$1$

Step 3

Differentiate the right side of the equation.

Tap for more steps...

Step 3.1

Convert from $\frac{1}{\sin (20 x)}$ to $\csc (20 x)$ .

$\frac{d}{d y} [\csc (20 x)]$

Step 3.2

Differentiate using the chain rule, which states that $\frac{d}{d y} [f (g (y))]$ is $f' (g (y)) g' (y)$ where $f (y) = \csc (y)$ and $g (y) = 20 x$ .

Tap for more steps...

Step 3.2.1

To apply the Chain Rule, set $u$ as $20 x$ .

$\frac{d}{d u} [\csc (u)] \frac{d}{d y} [20 x]$

Step 3.2.2

The derivative of $\csc (u)$ with respect to $u$ is $- \csc (u) \cot (u)$ .

$- \csc (u) \cot (u) \frac{d}{d y} [20 x]$

Step 3.2.3

Replace all occurrences of $u$ with $20 x$ .

$- \csc (20 x) \cot (20 x) \frac{d}{d y} [20 x]$

Step 3.3

Differentiate using the Constant Multiple Rule.

Tap for more steps...

Step 3.3.1

Since $20$ is constant with respect to $y$ , the derivative of $20 x$ with respect to $y$ is $20 \frac{d}{d y} [x]$ .

$- \csc (20 x) \cot (20 x) (20 \frac{d}{d y} [x])$

Step 3.3.2

Multiply $20$ by $- 1$ .

$- 20 \csc (20 x) \cot (20 x) \frac{d}{d y} [x]$

Step 3.4

Rewrite $\frac{d}{d y} [x]$ as $x'$ .

$- 20 \csc (20 x) \cot (20 x) x'$

Step 3.5

Reorder the factors of $- 20 \csc (20 x) \cot (20 x) x'$ .

$- 20 \cot (20 x) \csc (20 x) x'$

Step 4

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

$1 = - 20 \cot (20 x) \csc (20 x) x'$

Step 5

Solve for $x'$ .

Tap for more steps...

Step 5.1

Rewrite the equation as $- 20 \cot (20 x) \csc (20 x) x' = 1$ .

$- 20 \cot (20 x) \csc (20 x) x' = 1$

Step 5.2

Divide each term in $- 20 \cot (20 x) \csc (20 x) x' = 1$ by $- 20 \cot (20 x) \csc (20 x)$ and simplify.

Tap for more steps...

Step 5.2.1

Divide each term in $- 20 \cot (20 x) \csc (20 x) x' = 1$ by $- 20 \cot (20 x) \csc (20 x)$ .

$\frac{- 20 \cot (20 x) \csc (20 x) x'}{- 20 \cot (20 x) \csc (20 x)} = \frac{1}{- 20 \cot (20 x) \csc (20 x)}$

Step 5.2.2

Simplify the left side.

Tap for more steps...

Step 5.2.2.1

Cancel the common factor of $- 20$ .

Tap for more steps...

Step 5.2.2.1.1

Cancel the common factor.

$\frac{- 20 \cot (20 x) \csc (20 x) x'}{- 20 \cot (20 x) \csc (20 x)} = \frac{1}{- 20 \cot (20 x) \csc (20 x)}$

Step 5.2.2.1.2

Rewrite the expression.

$\frac{\cot (20 x) \csc (20 x) x'}{\cot (20 x) \csc (20 x)} = \frac{1}{- 20 \cot (20 x) \csc (20 x)}$

Step 5.2.2.2

Cancel the common factor of $\cot (20 x)$ .

Tap for more steps...

Step 5.2.2.2.1

Cancel the common factor.

$\frac{\cot (20 x) \csc (20 x) x'}{\cot (20 x) \csc (20 x)} = \frac{1}{- 20 \cot (20 x) \csc (20 x)}$

Step 5.2.2.2.2

Rewrite the expression.

$\frac{\csc (20 x) x'}{\csc (20 x)} = \frac{1}{- 20 \cot (20 x) \csc (20 x)}$

Step 5.2.2.3

Cancel the common factor of $\csc (20 x)$ .

Tap for more steps...

Step 5.2.2.3.1

Cancel the common factor.

$\frac{\csc (20 x) x'}{\csc (20 x)} = \frac{1}{- 20 \cot (20 x) \csc (20 x)}$

Step 5.2.2.3.2

Divide $x'$ by $1$ .

$x' = \frac{1}{- 20 \cot (20 x) \csc (20 x)}$

Step 5.2.3

Simplify the right side.

Tap for more steps...

Step 5.2.3.1

Separate fractions.

$x' = \frac{1}{- 20 \csc (20 x)} \cdot \frac{1}{\cot (20 x)}$

Step 5.2.3.2

Rewrite $\cot (20 x)$ in terms of sines and cosines.

$x' = \frac{1}{- 20 \csc (20 x)} \cdot \frac{1}{\frac{\cos (20 x)}{\sin (20 x)}}$

Step 5.2.3.3

Multiply by the reciprocal of the fraction to divide by $\frac{\cos (20 x)}{\sin (20 x)}$ .

$x' = \frac{1}{- 20 \csc (20 x)} \cdot \frac{\sin (20 x)}{\cos (20 x)}$

Step 5.2.3.4

Convert from $\frac{\sin (20 x)}{\cos (20 x)}$ to $\tan (20 x)$ .

$x' = \frac{1}{- 20 \csc (20 x)} \tan (20 x)$

Step 5.2.3.5

Separate fractions.

$x' = \frac{1}{- 20} \cdot \frac{1}{\csc (20 x)} \tan (20 x)$

Step 5.2.3.6

Rewrite $\csc (20 x)$ in terms of sines and cosines.

$x' = \frac{1}{- 20} \cdot \frac{1}{\frac{1}{\sin (20 x)}} \tan (20 x)$

Step 5.2.3.7

Multiply by the reciprocal of the fraction to divide by $\frac{1}{\sin (20 x)}$ .

$x' = \frac{1}{- 20} (1 \sin (20 x)) \tan (20 x)$

Step 5.2.3.8

Simplify the expression.

Tap for more steps...

Step 5.2.3.8.1

Multiply $\sin (20 x)$ by $1$ .

$x' = \frac{1}{- 20} \sin (20 x) \tan (20 x)$

Step 5.2.3.8.2

Move the negative in front of the fraction.

$x' = - \frac{1}{20} \sin (20 x) \tan (20 x)$

Step 5.2.3.9

Combine $\sin (20 x)$ and $\frac{1}{20}$ .

$x' = - \frac{\sin (20 x)}{20} \tan (20 x)$

Step 5.2.3.10

Combine $\tan (20 x)$ and $\frac{\sin (20 x)}{20}$ .

$x' = - \frac{\tan (20 x) \sin (20 x)}{20}$

Step 6

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

$\frac{d x}{d y} = - \frac{\tan (20 x) \sin (20 x)}{20}$