Calculus Examples

$\frac{y}{1 - y}$

Step 1

Differentiate using the Quotient Rule which states that $\frac{d}{d y} [\frac{f (y)}{g (y)}]$ is $\frac{g (y) \frac{d}{d y} [f (y)] - f (y) \frac{d}{d y} [g (y)]}{{g (y)}^{2}}$ where $f (y) = y$ and $g (y) = 1 - y$ .

$\frac{(1 - y) \frac{d}{d y} [y] - y \frac{d}{d y} [1 - y]}{{(1 - y)}^{2}}$

Step 2

Differentiate.

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Step 2.1

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

$\frac{(1 - y) \cdot 1 - y \frac{d}{d y} [1 - y]}{{(1 - y)}^{2}}$

Step 2.2

Multiply $1 - y$ by $1$ .

$\frac{1 - y - y \frac{d}{d y} [1 - y]}{{(1 - y)}^{2}}$

Step 2.3

By the Sum Rule, the derivative of $1 - y$ with respect to $y$ is $\frac{d}{d y} [1] + \frac{d}{d y} [- y]$ .

$\frac{1 - y - y (\frac{d}{d y} [1] + \frac{d}{d y} [- y])}{{(1 - y)}^{2}}$

Step 2.4

Since $1$ is constant with respect to $y$ , the derivative of $1$ with respect to $y$ is $0$ .

$\frac{1 - y - y (0 + \frac{d}{d y} [- y])}{{(1 - y)}^{2}}$

Step 2.5

Add $0$ and $\frac{d}{d y} [- y]$ .

$\frac{1 - y - y \frac{d}{d y} [- y]}{{(1 - y)}^{2}}$

Step 2.6

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

$\frac{1 - y - y (- \frac{d}{d y} [y])}{{(1 - y)}^{2}}$

Step 2.7

Multiply.

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Step 2.7.1

Multiply $- 1$ by $- 1$ .

$\frac{1 - y + 1 y \frac{d}{d y} [y]}{{(1 - y)}^{2}}$

Step 2.7.2

Multiply $y$ by $1$ .

$\frac{1 - y + y \frac{d}{d y} [y]}{{(1 - y)}^{2}}$

Step 2.8

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

$\frac{1 - y + y \cdot 1}{{(1 - y)}^{2}}$

Step 2.9

Simplify by adding terms.

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Step 2.9.1

Multiply $y$ by $1$ .

$\frac{1 - y + y}{{(1 - y)}^{2}}$

Step 2.9.2

Add $- y$ and $y$ .

$\frac{1 + 0}{{(1 - y)}^{2}}$

Step 2.9.3

Simplify the expression.

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Step 2.9.3.1

Add $1$ and $0$ .

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

Step 2.9.3.2

Reorder terms.

$\frac{1}{{(- y + 1)}^{2}}$