Calculus Examples

$- \frac{y}{x + 2 y}$

Step 1

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

$- \frac{d}{d y} [\frac{y}{x + 2 y}] + \frac{y}{x + 2 y} \frac{d}{d y} [- 1]$

Step 2

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 $g (y) = x + 2 y$ .

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

Step 3

Differentiate.

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Step 3.1

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

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

Step 3.2

Multiply $x + 2 y$ by $1$ .

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

Step 3.3

By the Sum Rule, the derivative of $x + 2 y$ with respect to $y$ is $\frac{d}{d y} [x] + \frac{d}{d y} [2 y]$ .

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

Step 3.4

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

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

Step 3.5

Add $0$ and $\frac{d}{d y} [2 y]$ .

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

Step 3.6

Since $2$ is constant with respect to $y$ , the derivative of $2 y$ with respect to $y$ is $2 \frac{d}{d y} [y]$ .

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

Step 3.7

Multiply $2$ by $- 1$ .

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

Step 3.8

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

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

Step 3.9

Simplify by adding terms.

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Step 3.9.1

Multiply $- 2$ by $1$ .

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

Step 3.9.2

Subtract $2 y$ from $2 y$ .

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

Step 3.9.3

Add $x$ and $0$ .

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

Step 3.10

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

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

Step 3.11

Simplify the expression.

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Step 3.11.1

Multiply $\frac{y}{x + 2 y}$ by $0$ .

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

Step 3.11.2

Add $- \frac{x}{{(x + 2 y)}^{2}}$ and $0$ .

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