Calculus Examples

$y = \frac{x}{x + y}$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (y) = \frac{d}{d x} (\frac{x}{x + y})$

Step 2

The derivative of $y$ with respect to $x$ is $y'$ .

$y'$

Step 3

Differentiate the right side of the equation.

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Step 3.1

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

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

Step 3.2

Differentiate.

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Step 3.2.1

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

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

Step 3.2.2

Multiply $x + y$ by $1$ .

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

Step 3.2.3

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

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

Step 3.2.4

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

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

Step 3.3

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

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

Step 3.4

Simplify.

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Step 3.4.1

Apply the distributive property.

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

Step 3.4.2

Simplify the numerator.

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Step 3.4.2.1

Multiply $- 1$ by $1$ .

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

Step 3.4.2.2

Combine the opposite terms in $x + y - x - x y'$ .

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Step 3.4.2.2.1

Subtract $x$ from $x$ .

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

Step 3.4.2.2.2

Add $y$ and $0$ .

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

Step 3.4.3

Reorder terms.

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

Step 3.4.4

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

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

Step 3.4.5

Factor $- 1$ out of $y$ .

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

Step 3.4.6

Factor $- 1$ out of $- (x y') - 1 (- y)$ .

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

Step 3.4.7

Rewrite $- (x y' - y)$ as $- 1 (x y' - y)$ .

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

Step 3.4.8

Move the negative in front of the fraction.

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

Step 4

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

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

Step 5

Solve for $y'$ .

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Step 5.1

Multiply both sides by ${(x + y)}^{2}$ .

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

Step 5.2

Simplify the right side.

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Step 5.2.1

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

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Step 5.2.1.1

Cancel the common factor of ${(x + y)}^{2}$ .

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Step 5.2.1.1.1

Move the leading negative in $- \frac{x y' - y}{{(x + y)}^{2}}$ into the numerator.

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

Step 5.2.1.1.2

Cancel the common factor.

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

Step 5.2.1.1.3

Rewrite the expression.

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

Step 5.2.1.2

Apply the distributive property.

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

Step 5.2.1.3

Multiply $- - y$ .

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Step 5.2.1.3.1

Multiply $- 1$ by $- 1$ .

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

Step 5.2.1.3.2

Multiply $y$ by $1$ .

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

Step 5.2.1.4

Move $x$ .

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

Step 5.3

Solve for $y'$ .

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Step 5.3.1

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

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

Step 5.3.2

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

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Step 5.3.2.1

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

$y' {(x + y)}^{2} + y' x = y$

Step 5.3.2.2

Simplify each term.

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Step 5.3.2.2.1

Rewrite ${(x + y)}^{2}$ as $(x + y) (x + y)$ .

$y' ((x + y) (x + y)) + y' (x) = y$

Step 5.3.2.2.2

Expand $(x + y) (x + y)$ using the FOIL Method.

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Step 5.3.2.2.2.1

Apply the distributive property.

$y' (x (x + y) + y (x + y)) + y' (x) = y$

Step 5.3.2.2.2.2

Apply the distributive property.

$y' (x \cdot x + x y + y (x + y)) + y' (x) = y$

Step 5.3.2.2.2.3

Apply the distributive property.

$y' (x \cdot x + x y + y x + y \cdot y) + y' (x) = y$

Step 5.3.2.2.3

Simplify and combine like terms.

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Step 5.3.2.2.3.1

Simplify each term.

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Step 5.3.2.2.3.1.1

Multiply $x$ by $x$ .

$y' (x^{2} + x y + y x + y \cdot y) + y' (x) = y$

Step 5.3.2.2.3.1.2

Multiply $y$ by $y$ .

$y' (x^{2} + x y + y x + y^{2}) + y' (x) = y$

Step 5.3.2.2.3.2

Add $x y$ and $y x$ .

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Step 5.3.2.2.3.2.1

Reorder $y$ and $x$ .

$y' (x^{2} + x y + x y + y^{2}) + y' (x) = y$

Step 5.3.2.2.3.2.2

Add $x y$ and $x y$ .

$y' (x^{2} + 2 x y + y^{2}) + y' (x) = y$

Step 5.3.2.2.4

Apply the distributive property.

$y' (x^{2}) + y' (2 x y) + y' (y^{2}) + y' (x) = y$

Step 5.3.2.2.5

Rewrite using the commutative property of multiplication.

$y' x^{2} + 2 y' x y + y' y^{2} + y' x = y$

Step 5.3.3

Factor $y'$ out of $y' x^{2} + 2 y' x y + y' y^{2} + y' x$ .

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Step 5.3.3.1

Factor $y'$ out of $y' x^{2}$ .

$y' (x^{2}) + 2 y' x y + y' (y^{2}) + y' (x) = y$

Step 5.3.3.2

Factor $y'$ out of $2 y' x y$ .

$y' (x^{2}) + y' (2 x y) + y' (y^{2}) + y' (x) = y$

Step 5.3.3.3

Factor $y'$ out of $y' y^{2}$ .

$y' (x^{2}) + y' (2 x y) + y' (y^{2}) + y' (x) = y$

Step 5.3.3.4

Factor $y'$ out of $y' x$ .

$y' (x^{2}) + y' (2 x y) + y' (y^{2}) + y' (x) = y$

Step 5.3.3.5

Factor $y'$ out of $y' (x^{2}) + y' (2 x y)$ .

$y' (x^{2} + 2 x y) + y' (y^{2}) + y' (x) = y$

Step 5.3.3.6

Factor $y'$ out of $y' (x^{2} + 2 x y) + y' (y^{2})$ .

$y' (x^{2} + 2 x y + y^{2}) + y' (x) = y$

Step 5.3.3.7

Factor $y'$ out of $y' (x^{2} + 2 x y + y^{2}) + y' (x)$ .

$y' (x^{2} + 2 x y + y^{2} + x) = y$

Step 5.3.4

Divide each term in $y' (x^{2} + 2 x y + y^{2} + x) = y$ by $x^{2} + 2 x y + y^{2} + x$ and simplify.

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Step 5.3.4.1

Divide each term in $y' (x^{2} + 2 x y + y^{2} + x) = y$ by $x^{2} + 2 x y + y^{2} + x$ .

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

Step 5.3.4.2

Simplify the left side.

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Step 5.3.4.2.1

Cancel the common factor of $x^{2} + 2 x y + y^{2} + x$ .

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Step 5.3.4.2.1.1

Cancel the common factor.

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

Step 5.3.4.2.1.2

Divide $y'$ by $1$ .

$y' = \frac{y}{x^{2} + 2 x y + y^{2} + x}$

Step 6

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

$\frac{d y}{d x} = \frac{y}{x^{2} + 2 x y + y^{2} + x}$