Calculus Examples

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

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 $g (y) = \frac{2 x + y}{x + 2 y}$ .

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

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

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

Step 3

Differentiate.

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Step 3.1

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

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

Step 3.2

Since $2 x$ is constant with respect to $y$ , the derivative of $2 x$ with respect to $y$ is $0$ .

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

Step 3.3

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

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

Step 3.4

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

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

Step 3.5

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

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

Step 3.6

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

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

Step 3.7

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

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

Step 3.8

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

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

Step 3.9

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

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

Step 3.10

Multiply $2$ by $- 1$ .

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

Step 3.11

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

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

Step 3.12

Multiply $- 2$ by $1$ .

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

Step 3.13

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

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

Step 3.14

Simplify the expression.

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Step 3.14.1

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

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

Step 3.14.2

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

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

Step 4

Simplify.

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Step 4.1

Apply the distributive property.

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

Step 4.2

Simplify the numerator.

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Step 4.2.1

Combine the opposite terms in $x + 2 y - 2 (2 x) - 2 y$ .

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Step 4.2.1.1

Subtract $2 y$ from $2 y$ .

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

Step 4.2.1.2

Add $x - 2 (2 x)$ and $0$ .

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

Step 4.2.2

Multiply $2$ by $- 2$ .

$- \frac{x - 4 x}{{(x + 2 y)}^{2}}$

Step 4.2.3

Subtract $4 x$ from $x$ .

$- \frac{- 3 x}{{(x + 2 y)}^{2}}$

Step 4.3

Combine terms.

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Step 4.3.1

Move the negative in front of the fraction.

$- - \frac{3 x}{{(x + 2 y)}^{2}}$

Step 4.3.2

Multiply $- 1$ by $- 1$ .

$1 \frac{3 x}{{(x + 2 y)}^{2}}$

Step 4.3.3

Multiply $\frac{3 x}{{(x + 2 y)}^{2}}$ by $1$ .

$\frac{3 x}{{(x + 2 y)}^{2}}$