Calculus Examples

$\frac{x^{2}}{x + y} = y^{2} + 8$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (\frac{x^{2}}{x + y}) = \frac{d}{d x} (y^{2} + 8)$

Step 2

Differentiate the left side of the equation.

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Step 2.1

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

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

Step 2.2

Differentiate.

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Step 2.2.1

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

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

Step 2.2.2

Move $2$ to the left of $x + y$ .

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

Step 2.2.3

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

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

Step 2.2.4

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

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

Step 2.3

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

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

Step 2.4

Simplify.

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Step 2.4.1

Apply the distributive property.

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

Step 2.4.2

Apply the distributive property.

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

Step 2.4.3

Apply the distributive property.

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

Step 2.4.4

Simplify the numerator.

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Step 2.4.4.1

Simplify each term.

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Step 2.4.4.1.1

Multiply $x$ by $x$ by adding the exponents.

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Step 2.4.4.1.1.1

Move $x$ .

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

Step 2.4.4.1.1.2

Multiply $x$ by $x$ .

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

Step 2.4.4.1.2

Multiply $- 1$ by $1$ .

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

Step 2.4.4.2

Subtract $x^{2}$ from $2 x^{2}$ .

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

Step 2.4.5

Reorder terms.

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

Step 2.4.6

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

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Step 2.4.6.1

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

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

Step 2.4.6.2

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

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

Step 2.4.6.3

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

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

Step 2.4.6.4

Factor $x$ out of $x \cdot x + x (2 y)$ .

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

Step 2.4.6.5

Factor $x$ out of $x (x + 2 y) + x (- x y')$ .

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

Step 3

Differentiate the right side of the equation.

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Step 3.1

By the Sum Rule, the derivative of $y^{2} + 8$ with respect to $x$ is $\frac{d}{d x} [y^{2}] + \frac{d}{d x} [8]$ .

$\frac{d}{d x} [y^{2}] + \frac{d}{d x} [8]$

Step 3.2

Evaluate $\frac{d}{d x} [y^{2}]$ .

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Step 3.2.1

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{2}$ and $g (x) = y$ .

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Step 3.2.1.1

To apply the Chain Rule, set $u$ as $y$ .

$\frac{d}{d u} [u^{2}] \frac{d}{d x} [y] + \frac{d}{d x} [8]$

Step 3.2.1.2

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = 2$ .

$2 u \frac{d}{d x} [y] + \frac{d}{d x} [8]$

Step 3.2.1.3

Replace all occurrences of $u$ with $y$ .

$2 y \frac{d}{d x} [y] + \frac{d}{d x} [8]$

Step 3.2.2

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

$2 y y' + \frac{d}{d x} [8]$

Step 3.3

Differentiate using the Constant Rule.

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Step 3.3.1

Since $8$ is constant with respect to $x$ , the derivative of $8$ with respect to $x$ is $0$ .

$2 y y' + 0$

Step 3.3.2

Add $2 y y'$ and $0$ .

$2 y y'$

Step 4

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

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

Step 5

Solve for $y'$ .

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Step 5.1

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

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

Step 5.2

Simplify the left side.

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Step 5.2.1

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

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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

Cancel the common factor.

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

Step 5.2.1.1.2

Rewrite the expression.

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

Step 5.2.1.2

Apply the distributive property.

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

Step 5.2.1.3

Simplify.

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Step 5.2.1.3.1

Multiply $x$ by $x$ .

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

Step 5.2.1.3.2

Rewrite using the commutative property of multiplication.

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

Step 5.2.1.3.3

Rewrite using the commutative property of multiplication.

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

Step 5.2.1.4

Multiply $x$ by $x$ by adding the exponents.

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Step 5.2.1.4.1

Move $x$ .

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

Step 5.2.1.4.2

Multiply $x$ by $x$ .

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

Step 5.2.1.5

Move $x^{2}$ .

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

Step 5.2.1.6

Move $x^{2}$ .

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

Step 5.2.1.7

Reorder $2 x y$ and $- y' x^{2}$ .

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

Step 5.3

Solve for $y'$ .

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Step 5.3.1

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

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Step 5.3.1.1

Rewrite.

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

Step 5.3.1.2

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

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

Step 5.3.1.3

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

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Step 5.3.1.3.1

Apply the distributive property.

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

Step 5.3.1.3.2

Apply the distributive property.

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

Step 5.3.1.3.3

Apply the distributive property.

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

Step 5.3.1.4

Simplify and combine like terms.

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Step 5.3.1.4.1

Simplify each term.

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Step 5.3.1.4.1.1

Multiply $x$ by $x$ .

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

Step 5.3.1.4.1.2

Multiply $y$ by $y$ .

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

Step 5.3.1.4.2

Add $x y$ and $y x$ .

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Step 5.3.1.4.2.1

Reorder $y$ and $x$ .

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

Step 5.3.1.4.2.2

Add $x y$ and $x y$ .

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

Step 5.3.1.5

Apply the distributive property.

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

Step 5.3.1.6

Simplify.

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Step 5.3.1.6.1

Multiply $y$ by $y$ by adding the exponents.

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Step 5.3.1.6.1.1

Move $y$ .

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

Step 5.3.1.6.1.2

Multiply $y$ by $y$ .

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

Step 5.3.1.6.2

Multiply $y$ by $y^{2}$ by adding the exponents.

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Step 5.3.1.6.2.1

Move $y^{2}$ .

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

Step 5.3.1.6.2.2

Multiply $y^{2}$ by $y$ .

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Step 5.3.1.6.2.2.1

Raise $y$ to the power of $1$ .

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

Step 5.3.1.6.2.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 5.3.1.6.2.3

Add $2$ and $1$ .

$- y' x^{2} + 2 x y + x^{2} = 2 y y' x^{2} + 2 y^{2} y' (2 x) + 2 y^{3} y'$

Step 5.3.1.7

Multiply $2$ by $2$ .

$- y' x^{2} + 2 x y + x^{2} = 2 y y' x^{2} + 4 y^{2} y' x + 2 y^{3} y'$

Step 5.3.2

Since $y'$ is on the right side of the equation, switch the sides so it is on the left side of the equation.

$2 y y' x^{2} + 4 y^{2} y' x + 2 y^{3} y' = - y' x^{2} + 2 x y + x^{2}$

Step 5.3.3

Add $y' x^{2}$ to both sides of the equation.

$2 y y' x^{2} + 4 y^{2} y' x + 2 y^{3} y' + y' x^{2} = 2 x y + x^{2}$

Step 5.3.4

Factor $y'$ out of $2 y y' x^{2} + 4 y^{2} y' x + 2 y^{3} y' + y' x^{2}$ .

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Step 5.3.4.1

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

$y' (2 y x^{2}) + 4 y^{2} y' x + 2 y^{3} y' + y' (x^{2}) = 2 x y + x^{2}$

Step 5.3.4.2

Factor $y'$ out of $4 y^{2} y' x$ .

$y' (2 y x^{2}) + y' (4 y^{2} x) + 2 y^{3} y' + y' (x^{2}) = 2 x y + x^{2}$

Step 5.3.4.3

Factor $y'$ out of $2 y^{3} y'$ .

$y' (2 y x^{2}) + y' (4 y^{2} x) + y' (2 y^{3}) + y' (x^{2}) = 2 x y + x^{2}$

Step 5.3.4.4

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

$y' (2 y x^{2}) + y' (4 y^{2} x) + y' (2 y^{3}) + y' (x^{2}) = 2 x y + x^{2}$

Step 5.3.4.5

Factor $y'$ out of $y' (2 y x^{2}) + y' (4 y^{2} x)$ .

$y' (2 y x^{2} + 4 y^{2} x) + y' (2 y^{3}) + y' (x^{2}) = 2 x y + x^{2}$

Step 5.3.4.6

Factor $y'$ out of $y' (2 y x^{2} + 4 y^{2} x) + y' (2 y^{3})$ .

$y' (2 y x^{2} + 4 y^{2} x + 2 y^{3}) + y' (x^{2}) = 2 x y + x^{2}$

Step 5.3.4.7

Factor $y'$ out of $y' (2 y x^{2} + 4 y^{2} x + 2 y^{3}) + y' (x^{2})$ .

$y' (2 y x^{2} + 4 y^{2} x + 2 y^{3} + x^{2}) = 2 x y + x^{2}$

Step 5.3.5

Divide each term in $y' (2 y x^{2} + 4 y^{2} x + 2 y^{3} + x^{2}) = 2 x y + x^{2}$ by $2 y x^{2} + 4 y^{2} x + 2 y^{3} + x^{2}$ and simplify.

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Step 5.3.5.1

Divide each term in $y' (2 y x^{2} + 4 y^{2} x + 2 y^{3} + x^{2}) = 2 x y + x^{2}$ by $2 y x^{2} + 4 y^{2} x + 2 y^{3} + x^{2}$ .

$\frac{y' (2 y x^{2} + 4 y^{2} x + 2 y^{3} + x^{2})}{2 y x^{2} + 4 y^{2} x + 2 y^{3} + x^{2}} = \frac{2 x y}{2 y x^{2} + 4 y^{2} x + 2 y^{3} + x^{2}} + \frac{x^{2}}{2 y x^{2} + 4 y^{2} x + 2 y^{3} + x^{2}}$

Step 5.3.5.2

Simplify the left side.

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Step 5.3.5.2.1

Cancel the common factor of $2 y x^{2} + 4 y^{2} x + 2 y^{3} + x^{2}$ .

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Step 5.3.5.2.1.1

Cancel the common factor.

Step 5.3.5.2.1.2

Divide $y'$ by $1$ .

$y' = \frac{2 x y}{2 y x^{2} + 4 y^{2} x + 2 y^{3} + x^{2}} + \frac{x^{2}}{2 y x^{2} + 4 y^{2} x + 2 y^{3} + x^{2}}$

Step 5.3.5.3

Simplify the right side.

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Step 5.3.5.3.1

Combine the numerators over the common denominator.

$y' = \frac{2 x y + x^{2}}{2 y x^{2} + 4 y^{2} x + 2 y^{3} + x^{2}}$

Step 5.3.5.3.2

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

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Step 5.3.5.3.2.1

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

$y' = \frac{x (2 y) + x^{2}}{2 y x^{2} + 4 y^{2} x + 2 y^{3} + x^{2}}$

Step 5.3.5.3.2.2

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

$y' = \frac{x (2 y) + x \cdot x}{2 y x^{2} + 4 y^{2} x + 2 y^{3} + x^{2}}$

Step 5.3.5.3.2.3

Factor $x$ out of $x (2 y) + x \cdot x$ .

$y' = \frac{x (2 y + x)}{2 y x^{2} + 4 y^{2} x + 2 y^{3} + x^{2}}$

Step 6

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

$\frac{d y}{d x} = \frac{x (2 y + x)}{2 y x^{2} + 4 y^{2} x + 2 y^{3} + x^{2}}$