Calculus Examples

$\cot (y) = x - y$

Step 1

Differentiate both sides of the equation.

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

Step 2

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

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

Step 3

Differentiate the right side of the equation.

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Step 3.1

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

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

Step 3.2

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

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

Step 3.3

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

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Step 3.3.1

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

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

Step 3.3.2

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

$x' - 1 \cdot 1$

Step 3.3.3

Multiply $- 1$ by $1$ .

$x' - 1$

Step 4

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

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

Step 5

Solve for $x'$ .

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Step 5.1

Rewrite the equation as $x' - 1 = - \csc^{2} (y)$ .

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

Step 5.2

Move all terms not containing $x'$ to the right side of the equation.

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Step 5.2.1

Add $1$ to both sides of the equation.

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

Step 5.2.2

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

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

Step 5.2.3

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

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

Step 5.2.4

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

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

Step 5.2.5

Apply pythagorean identity.

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

Step 6

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

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