Calculus Examples

$\cot (y) = x - y$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (\cot (y)) = \frac{d}{d x} (x - y)$

Step 2

Differentiate the left side of the equation.

Tap for more steps...

Step 2.1

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

Tap for more steps...

Step 2.1.1

To apply the Chain Rule, set $u$ as $y$ .

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

Step 2.1.2

The derivative of $\cot (u)$ with respect to $u$ is $- \csc^{2} (u)$ .

$- \csc^{2} (u) \frac{d}{d x} [y]$

Step 2.1.3

Replace all occurrences of $u$ with $y$ .

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

Step 2.2

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

$- \csc^{2} (y) y'$

Step 3

Differentiate the right side of the equation.

Tap for more steps...

Step 3.1

Differentiate.

Tap for more steps...

Step 3.1.1

By the Sum Rule, the derivative of $x - y$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [- y]$ .

$\frac{d}{d x} [x] + \frac{d}{d x} [- y]$

Step 3.1.2

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

$1 + \frac{d}{d x} [- y]$

Step 3.2

Evaluate $\frac{d}{d x} [- y]$ .

Tap for more steps...

Step 3.2.1

Since $- 1$ is constant with respect to $x$ , the derivative of $- y$ with respect to $x$ is $- \frac{d}{d x} [y]$ .

$1 - \frac{d}{d x} [y]$

Step 3.2.2

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

$1 - y'$

Step 4

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

$- \csc^{2} (y) y' = 1 - y'$

Step 5

Solve for $y'$ .

Tap for more steps...

Step 5.1

Simplify the left side.

Tap for more steps...

Step 5.1.1

Reorder factors in $- \csc^{2} (y) y'$ .

$- y' \csc^{2} (y) = 1 - y'$

Step 5.2

Move all terms containing $y'$ to the left side of the equation.

Tap for more steps...

Step 5.2.1

Add $y'$ to both sides of the equation.

$- y' \csc^{2} (y) + y' = 1$

Step 5.2.2

Reorder $- y' \csc^{2} (y)$ and $y'$ .

$y' - y' \csc^{2} (y) = 1$

Step 5.2.3

Rewrite $y'$ as $1 y'$ .

$1 y' - y' \csc^{2} (y) = 1$

Step 5.2.4

Factor $- y'$ out of $1 y'$ .

$- y' \cdot - 1 - y' \csc^{2} (y) = 1$

Step 5.2.5

Factor $- y'$ out of $- y' \csc^{2} (y)$ .

$- y' \cdot - 1 - y' \csc^{2} (y) = 1$

Step 5.2.6

Factor $- y'$ out of $- y' \cdot - 1 - y' \csc^{2} (y)$ .

$- y' (- 1 + \csc^{2} (y)) = 1$

Step 5.2.7

Rearrange terms.

$- y' (\csc^{2} (y) - 1) = 1$

Step 5.2.8

Apply pythagorean identity.

$- y' \cot^{2} (y) = 1$

Step 5.3

Divide each term in $- y' \cot^{2} (y) = 1$ by $- \cot^{2} (y)$ and simplify.

Tap for more steps...

Step 5.3.1

Divide each term in $- y' \cot^{2} (y) = 1$ by $- \cot^{2} (y)$ .

$\frac{- y' \cot^{2} (y)}{- \cot^{2} (y)} = \frac{1}{- \cot^{2} (y)}$

Step 5.3.2

Simplify the left side.

Tap for more steps...

Step 5.3.2.1

Dividing two negative values results in a positive value.

$\frac{y' \cot^{2} (y)}{\cot^{2} (y)} = \frac{1}{- \cot^{2} (y)}$

Step 5.3.2.2

Cancel the common factor of $\cot^{2} (y)$ .

Tap for more steps...

Step 5.3.2.2.1

Cancel the common factor.

$\frac{y' \cot^{2} (y)}{\cot^{2} (y)} = \frac{1}{- \cot^{2} (y)}$

Step 5.3.2.2.2

Divide $y'$ by $1$ .

$y' = \frac{1}{- \cot^{2} (y)}$

Step 5.3.3

Simplify the right side.

Tap for more steps...

Step 5.3.3.1

Cancel the common factor of $1$ and $- 1$ .

Tap for more steps...

Step 5.3.3.1.1

Rewrite $1$ as $- 1 (- 1)$ .

$y' = \frac{- 1 (- 1)}{- \cot^{2} (y)}$

Step 5.3.3.1.2

Move the negative in front of the fraction.

$y' = - \frac{1}{\cot^{2} (y)}$

Step 5.3.3.2

Rewrite $1$ as $1^{2}$ .

$y' = - \frac{1^{2}}{\cot^{2} (y)}$

Step 5.3.3.3

Rewrite $\frac{1^{2}}{\cot^{2} (y)}$ as ${(\frac{1}{\cot (y)})}^{2}$ .

$y' = - {(\frac{1}{\cot (y)})}^{2}$

Step 5.3.3.4

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

$y' = - {(\frac{1}{\frac{\cos (y)}{\sin (y)}})}^{2}$

Step 5.3.3.5

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

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

Step 5.3.3.6

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

$y' = - \tan^{2} (y)$

Step 6

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

$\frac{d y}{d x} = - \tan^{2} (y)$