Calculus Examples

${(x + y)}^{3} = x^{3} + y^{3}$

Step 1

Differentiate both sides of the equation.

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

Step 2

Differentiate the left side of the equation.

Tap for more steps...

Step 2.1

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{3}$ and $g (x) = x + y$ .

Tap for more steps...

Step 2.1.1

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

$\frac{d}{d u} [u^{3}] \frac{d}{d x} [x + y]$

Step 2.1.2

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = 3$ .

$3 u^{2} \frac{d}{d x} [x + y]$

Step 2.1.3

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

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

Step 2.2

Differentiate.

Tap for more steps...

Step 2.2.1

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

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

Step 2.2.2

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

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

Step 2.3

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

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

Step 2.4

Simplify.

Tap for more steps...

Step 2.4.1

Apply the distributive property.

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

Step 2.4.2

Multiply $3$ by $1$ .

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

Step 3

Differentiate the right side of the equation.

Tap for more steps...

Step 3.1

Differentiate.

Tap for more steps...

Step 3.1.1

By the Sum Rule, the derivative of $x^{3} + y^{3}$ with respect to $x$ is $\frac{d}{d x} [x^{3}] + \frac{d}{d x} [y^{3}]$ .

$\frac{d}{d x} [x^{3}] + \frac{d}{d x} [y^{3}]$

Step 3.1.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 3$ .

$3 x^{2} + \frac{d}{d x} [y^{3}]$

Step 3.2

Evaluate $\frac{d}{d x} [y^{3}]$ .

Tap for more steps...

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 $f (x) = x^{3}$ and $g (x) = y$ .

Tap for more steps...

Step 3.2.1.1

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

$3 x^{2} + \frac{d}{d u} [u^{3}] \frac{d}{d x} [y]$

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 $n = 3$ .

$3 x^{2} + 3 u^{2} \frac{d}{d x} [y]$

Step 3.2.1.3

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

$3 x^{2} + 3 y^{2} \frac{d}{d x} [y]$

Step 3.2.2

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

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

Step 4

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

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

Step 5

Solve for $y'$ .

Tap for more steps...

Step 5.1

Reorder factors in $3 {(x + y)}^{2} + 3 {(x + y)}^{2} y'$ .

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

Step 5.2

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

Tap for more steps...

Step 5.2.1

Subtract $3 y^{2} y'$ from both sides of the equation.

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

Step 5.2.2

Simplify each term.

Tap for more steps...

Step 5.2.2.1

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

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

Step 5.2.2.2

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

Tap for more steps...

Step 5.2.2.2.1

Apply the distributive property.

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

Step 5.2.2.2.2

Apply the distributive property.

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

Step 5.2.2.2.3

Apply the distributive property.

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

Step 5.2.2.3

Simplify and combine like terms.

Tap for more steps...

Step 5.2.2.3.1

Simplify each term.

Tap for more steps...

Step 5.2.2.3.1.1

Multiply $x$ by $x$ .

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

Step 5.2.2.3.1.2

Multiply $y$ by $y$ .

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

Step 5.2.2.3.2

Add $x y$ and $y x$ .

Tap for more steps...

Step 5.2.2.3.2.1

Reorder $y$ and $x$ .

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

Step 5.2.2.3.2.2

Add $x y$ and $x y$ .

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

Step 5.2.2.4

Apply the distributive property.

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

Step 5.2.2.5

Multiply $2$ by $3$ .

$3 x^{2} + 6 (x y) + 3 y^{2} + 3 y' {(x + y)}^{2} - 3 y^{2} y' = 3 x^{2}$

Step 5.2.2.6

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

$3 x^{2} + 6 x y + 3 y^{2} + 3 y' ((x + y) (x + y)) - 3 y^{2} y' = 3 x^{2}$

Step 5.2.2.7

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

Tap for more steps...

Step 5.2.2.7.1

Apply the distributive property.

$3 x^{2} + 6 x y + 3 y^{2} + 3 y' (x (x + y) + y (x + y)) - 3 y^{2} y' = 3 x^{2}$

Step 5.2.2.7.2

Apply the distributive property.

$3 x^{2} + 6 x y + 3 y^{2} + 3 y' (x \cdot x + x y + y (x + y)) - 3 y^{2} y' = 3 x^{2}$

Step 5.2.2.7.3

Apply the distributive property.

$3 x^{2} + 6 x y + 3 y^{2} + 3 y' (x \cdot x + x y + y x + y \cdot y) - 3 y^{2} y' = 3 x^{2}$

Step 5.2.2.8

Simplify and combine like terms.

Tap for more steps...

Step 5.2.2.8.1

Simplify each term.

Tap for more steps...

Step 5.2.2.8.1.1

Multiply $x$ by $x$ .

$3 x^{2} + 6 x y + 3 y^{2} + 3 y' (x^{2} + x y + y x + y \cdot y) - 3 y^{2} y' = 3 x^{2}$

Step 5.2.2.8.1.2

Multiply $y$ by $y$ .

$3 x^{2} + 6 x y + 3 y^{2} + 3 y' (x^{2} + x y + y x + y^{2}) - 3 y^{2} y' = 3 x^{2}$

Step 5.2.2.8.2

Add $x y$ and $y x$ .

Tap for more steps...

Step 5.2.2.8.2.1

Reorder $y$ and $x$ .

$3 x^{2} + 6 x y + 3 y^{2} + 3 y' (x^{2} + x y + x y + y^{2}) - 3 y^{2} y' = 3 x^{2}$

Step 5.2.2.8.2.2

Add $x y$ and $x y$ .

$3 x^{2} + 6 x y + 3 y^{2} + 3 y' (x^{2} + 2 x y + y^{2}) - 3 y^{2} y' = 3 x^{2}$

Step 5.2.2.9

Apply the distributive property.

$3 x^{2} + 6 x y + 3 y^{2} + 3 y' x^{2} + 3 y' (2 x y) + 3 y' y^{2} - 3 y^{2} y' = 3 x^{2}$

Step 5.2.2.10

Multiply $2$ by $3$ .

$3 x^{2} + 6 x y + 3 y^{2} + 3 y' x^{2} + 6 y' x y + 3 y' y^{2} - 3 y^{2} y' = 3 x^{2}$

Step 5.2.3

Combine the opposite terms in $3 x^{2} + 6 x y + 3 y^{2} + 3 y' x^{2} + 6 y' x y + 3 y' y^{2} - 3 y^{2} y'$ .

Tap for more steps...

Step 5.2.3.1

Reorder the factors in the terms $3 y' y^{2}$ and $- 3 y^{2} y'$ .

$3 x^{2} + 6 x y + 3 y^{2} + 3 y' x^{2} + 6 y' x y + 3 y^{2} y' - 3 y^{2} y' = 3 x^{2}$

Step 5.2.3.2

Subtract $3 y^{2} y'$ from $3 y^{2} y'$ .

$3 x^{2} + 6 x y + 3 y^{2} + 3 y' x^{2} + 6 y' x y + 0 = 3 x^{2}$

Step 5.2.3.3

Add $3 x^{2} + 6 x y + 3 y^{2} + 3 y' x^{2} + 6 y' x y$ and $0$ .

$3 x^{2} + 6 x y + 3 y^{2} + 3 y' x^{2} + 6 y' x y = 3 x^{2}$

Step 5.3

Move all terms not containing $y'$ to the right side of the equation.

Tap for more steps...

Step 5.3.1

Subtract $3 x^{2}$ from both sides of the equation.

$6 x y + 3 y^{2} + 3 y' x^{2} + 6 y' x y = 3 x^{2} - 3 x^{2}$

Step 5.3.2

Subtract $6 x y$ from both sides of the equation.

$3 y^{2} + 3 y' x^{2} + 6 y' x y = 3 x^{2} - 3 x^{2} - 6 x y$

Step 5.3.3

Subtract $3 y^{2}$ from both sides of the equation.

$3 y' x^{2} + 6 y' x y = 3 x^{2} - 3 x^{2} - 6 x y - 3 y^{2}$

Step 5.3.4

Subtract $3 x^{2}$ from $3 x^{2}$ .

$3 y' x^{2} + 6 y' x y = 0 - 6 x y - 3 y^{2}$

Step 5.3.5

Subtract $6 x y$ from $0$ .

$3 y' x^{2} + 6 y' x y = - 6 x y - 3 y^{2}$

Step 5.4

Factor $3 y' x$ out of $3 y' x^{2} + 6 y' x y$ .

Tap for more steps...

Step 5.4.1

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

$3 y' x \cdot x + 6 y' x y = - 6 x y - 3 y^{2}$

Step 5.4.2

Factor $3 y' x$ out of $6 y' x y$ .

$3 y' x \cdot x + 3 y' x (2 y) = - 6 x y - 3 y^{2}$

Step 5.4.3

Factor $3 y' x$ out of $3 y' x \cdot x + 3 y' x (2 y)$ .

$3 y' x (x + 2 y) = - 6 x y - 3 y^{2}$

Step 5.5

Divide each term in $3 y' x (x + 2 y) = - 6 x y - 3 y^{2}$ by $3 x (x + 2 y)$ and simplify.

Tap for more steps...

Step 5.5.1

Divide each term in $3 y' x (x + 2 y) = - 6 x y - 3 y^{2}$ by $3 x (x + 2 y)$ .

$\frac{3 y' x (x + 2 y)}{3 x (x + 2 y)} = \frac{- 6 x y}{3 x (x + 2 y)} + \frac{- 3 y^{2}}{3 x (x + 2 y)}$

Step 5.5.2

Simplify the left side.

Tap for more steps...

Step 5.5.2.1

Cancel the common factor of $3$ .

Tap for more steps...

Step 5.5.2.1.1

Cancel the common factor.

$\frac{3 y' x (x + 2 y)}{3 x (x + 2 y)} = \frac{- 6 x y}{3 x (x + 2 y)} + \frac{- 3 y^{2}}{3 x (x + 2 y)}$

Step 5.5.2.1.2

Rewrite the expression.

$\frac{y' x (x + 2 y)}{x (x + 2 y)} = \frac{- 6 x y}{3 x (x + 2 y)} + \frac{- 3 y^{2}}{3 x (x + 2 y)}$

Step 5.5.2.2

Cancel the common factor of $x$ .

Tap for more steps...

Step 5.5.2.2.1

Cancel the common factor.

$\frac{y' x (x + 2 y)}{x (x + 2 y)} = \frac{- 6 x y}{3 x (x + 2 y)} + \frac{- 3 y^{2}}{3 x (x + 2 y)}$

Step 5.5.2.2.2

Rewrite the expression.

$\frac{y' (x + 2 y)}{x + 2 y} = \frac{- 6 x y}{3 x (x + 2 y)} + \frac{- 3 y^{2}}{3 x (x + 2 y)}$

Step 5.5.2.3

Cancel the common factor of $x + 2 y$ .

Tap for more steps...

Step 5.5.2.3.1

Cancel the common factor.

$\frac{y' (x + 2 y)}{x + 2 y} = \frac{- 6 x y}{3 x (x + 2 y)} + \frac{- 3 y^{2}}{3 x (x + 2 y)}$

Step 5.5.2.3.2

Divide $y'$ by $1$ .

$y' = \frac{- 6 x y}{3 x (x + 2 y)} + \frac{- 3 y^{2}}{3 x (x + 2 y)}$

Step 5.5.3

Simplify the right side.

Tap for more steps...

Step 5.5.3.1

Simplify each term.

Tap for more steps...

Step 5.5.3.1.1

Cancel the common factor of $- 6$ and $3$ .

Tap for more steps...

Step 5.5.3.1.1.1

Factor $3$ out of $- 6 x y$ .

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

Step 5.5.3.1.1.2

Cancel the common factors.

Tap for more steps...

Step 5.5.3.1.1.2.1

Factor $3$ out of $3 x (x + 2 y)$ .

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

Step 5.5.3.1.1.2.2

Cancel the common factor.

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

Step 5.5.3.1.1.2.3

Rewrite the expression.

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

Step 5.5.3.1.2

Cancel the common factor of $x$ .

Tap for more steps...

Step 5.5.3.1.2.1

Cancel the common factor.

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

Step 5.5.3.1.2.2

Rewrite the expression.

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

Step 5.5.3.1.3

Move the negative in front of the fraction.

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

Step 5.5.3.1.4

Cancel the common factor of $- 3$ and $3$ .

Tap for more steps...

Step 5.5.3.1.4.1

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

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

Step 5.5.3.1.4.2

Cancel the common factors.

Tap for more steps...

Step 5.5.3.1.4.2.1

Factor $3$ out of $3 x (x + 2 y)$ .

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

Step 5.5.3.1.4.2.2

Cancel the common factor.

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

Step 5.5.3.1.4.2.3

Rewrite the expression.

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

Step 5.5.3.1.5

Move the negative in front of the fraction.

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

Step 5.5.3.2

To write $- \frac{2 y}{x + 2 y}$ as a fraction with a common denominator, multiply by $\frac{x}{x}$ .

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

Step 5.5.3.3

Write each expression with a common denominator of $(x + 2 y) x$ , by multiplying each by an appropriate factor of $1$ .

Tap for more steps...

Step 5.5.3.3.1

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

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

Step 5.5.3.3.2

Reorder the factors of $(x + 2 y) x$ .

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

Step 5.5.3.4

Combine the numerators over the common denominator.

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

Step 5.5.3.5

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

Tap for more steps...

Step 5.5.3.5.1

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

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

Step 5.5.3.5.2

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

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

Step 5.5.3.5.3

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

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

Step 5.5.3.6

Factor $- 1$ out of $- 2 x$ .

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

Step 5.5.3.7

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

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

Step 5.5.3.8

Factor $- 1$ out of $- (2 x) - (y)$ .

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

Step 5.5.3.9

Simplify the expression.

Tap for more steps...

Step 5.5.3.9.1

Rewrite $- (2 x + y)$ as $- 1 (2 x + y)$ .

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

Step 5.5.3.9.2

Move the negative in front of the fraction.

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

Step 6

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

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