Calculus Examples

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

Step 1

Differentiate both sides of the equation.

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

Step 2

Differentiate the left side of the equation.

Tap for more steps...

Step 2.1

By the Sum Rule, the derivative of $x^{2} y^{2} + x y^{3}$ with respect to $x$ is $\frac{d}{d x} [x^{2} y^{2}] + \frac{d}{d x} [x y^{3}]$ .

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

Step 2.2

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

Tap for more steps...

Step 2.2.1

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

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

Step 2.2.2

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 $g (x) = y$ .

Tap for more steps...

Step 2.2.2.1

To apply the Chain Rule, set $u_{1}$ as $y$ .

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

Step 2.2.2.2

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

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

Step 2.2.2.3

Replace all occurrences of $u_{1}$ with $y$ .

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

Step 2.2.3

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

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

Step 2.2.4

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

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

Step 2.2.5

Move $2$ to the left of $x^{2}$ .

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

Step 2.2.6

Move $2$ to the left of $y^{2}$ .

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

Step 2.3

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

Tap for more steps...

Step 2.3.1

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

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

Step 2.3.2

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 2.3.2.1

To apply the Chain Rule, set $u_{2}$ as $y$ .

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

Step 2.3.2.2

Differentiate using the Power Rule which states that $\frac{d}{d u_{2}} [{u_{2}}^{n}]$ is $n {u_{2}}^{n - 1}$ where $n = 3$ .

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

Step 2.3.2.3

Replace all occurrences of $u_{2}$ with $y$ .

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

Step 2.3.3

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

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

Step 2.3.4

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

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

Step 2.3.5

Move $3$ to the left of $x$ .

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

Step 2.3.6

Multiply $y^{3}$ by $1$ .

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

Step 2.4

Reorder terms.

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

Step 3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 4$ .

$4 x^{3}$

Step 4

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

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

Step 5

Solve for $y'$ .

Tap for more steps...

Step 5.1

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

Tap for more steps...

Step 5.1.1

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

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

Step 5.1.2

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

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

Step 5.2

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

Tap for more steps...

Step 5.2.1

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

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

Step 5.2.2

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

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

Step 5.2.3

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

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

Step 5.3

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

Tap for more steps...

Step 5.3.1

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

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

Step 5.3.2

Simplify the left side.

Tap for more steps...

Step 5.3.2.1

Cancel the common factor of $x$ .

Tap for more steps...

Step 5.3.2.1.1

Cancel the common factor.

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

Step 5.3.2.1.2

Rewrite the expression.

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

Step 5.3.2.2

Cancel the common factor of $y$ .

Tap for more steps...

Step 5.3.2.2.1

Cancel the common factor.

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

Step 5.3.2.2.2

Rewrite the expression.

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

Step 5.3.2.3

Cancel the common factor of $2 x + 3 y$ .

Tap for more steps...

Step 5.3.2.3.1

Cancel the common factor.

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

Step 5.3.2.3.2

Divide $y'$ by $1$ .

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

Step 5.3.3

Simplify the right side.

Tap for more steps...

Step 5.3.3.1

Simplify each term.

Tap for more steps...

Step 5.3.3.1.1

Cancel the common factor of $x^{3}$ and $x$ .

Tap for more steps...

Step 5.3.3.1.1.1

Factor $x$ out of $4 x^{3}$ .

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

Step 5.3.3.1.1.2

Cancel the common factors.

Tap for more steps...

Step 5.3.3.1.1.2.1

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

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

Step 5.3.3.1.1.2.2

Cancel the common factor.

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

Step 5.3.3.1.1.2.3

Rewrite the expression.

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

Step 5.3.3.1.2

Cancel the common factor of $y^{3}$ and $y$ .

Tap for more steps...

Step 5.3.3.1.2.1

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

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

Step 5.3.3.1.2.2

Cancel the common factors.

Tap for more steps...

Step 5.3.3.1.2.2.1

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

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

Step 5.3.3.1.2.2.2

Cancel the common factor.

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

Step 5.3.3.1.2.2.3

Rewrite the expression.

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

Step 5.3.3.1.3

Move the negative in front of the fraction.

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

Step 5.3.3.1.4

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

Tap for more steps...

Step 5.3.3.1.4.1

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

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

Step 5.3.3.1.4.2

Cancel the common factors.

Tap for more steps...

Step 5.3.3.1.4.2.1

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

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

Step 5.3.3.1.4.2.2

Cancel the common factor.

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

Step 5.3.3.1.4.2.3

Rewrite the expression.

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

Step 5.3.3.1.5

Cancel the common factor of $x$ .

Tap for more steps...

Step 5.3.3.1.5.1

Cancel the common factor.

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

Step 5.3.3.1.5.2

Rewrite the expression.

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

Step 5.3.3.1.6

Move the negative in front of the fraction.

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

Step 5.3.3.2

To write $- \frac{2 y}{2 x + 3 y}$ as a fraction with a common denominator, multiply by $\frac{y}{y}$ .

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

Step 5.3.3.3

Write each expression with a common denominator of $y (2 x + 3 y)$ , by multiplying each by an appropriate factor of $1$ .

Tap for more steps...

Step 5.3.3.3.1

Multiply $\frac{2 y}{2 x + 3 y}$ by $\frac{y}{y}$ .

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

Step 5.3.3.3.2

Reorder the factors of $(2 x + 3 y) y$ .

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

Step 5.3.3.4

Combine the numerators over the common denominator.

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

Step 5.3.3.5

Simplify the numerator.

Tap for more steps...

Step 5.3.3.5.1

Factor $2$ out of $4 x^{2} - 2 y \cdot y$ .

Tap for more steps...

Step 5.3.3.5.1.1

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

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

Step 5.3.3.5.1.2

Factor $2$ out of $- 2 y \cdot y$ .

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

Step 5.3.3.5.1.3

Factor $2$ out of $2 (2 x^{2}) + 2 (- y \cdot y)$ .

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

Step 5.3.3.5.2

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

Tap for more steps...

Step 5.3.3.5.2.1

Move $y$ .

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

Step 5.3.3.5.2.2

Multiply $y$ by $y$ .

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

Step 5.3.3.6

To write $- \frac{y^{2}}{x (2 x + 3 y)}$ as a fraction with a common denominator, multiply by $\frac{y}{y}$ .

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

Step 5.3.3.7

To write $\frac{2 (2 x^{2} - y^{2})}{y (2 x + 3 y)}$ as a fraction with a common denominator, multiply by $\frac{x}{x}$ .

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

Step 5.3.3.8

Write each expression with a common denominator of $x (2 x + 3 y) y$ , by multiplying each by an appropriate factor of $1$ .

Tap for more steps...

Step 5.3.3.8.1

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

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

Step 5.3.3.8.2

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

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

Step 5.3.3.8.3

Reorder the factors of $x (2 x + 3 y) y$ .

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

Step 5.3.3.8.4

Reorder the factors of $y (2 x + 3 y) x$ .

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

Step 5.3.3.9

Combine the numerators over the common denominator.

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

Step 5.3.3.10

Simplify the numerator.

Tap for more steps...

Step 5.3.3.10.1

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

Tap for more steps...

Step 5.3.3.10.1.1

Move $y$ .

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

Step 5.3.3.10.1.2

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

Tap for more steps...

Step 5.3.3.10.1.2.1

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

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

Step 5.3.3.10.1.2.2

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

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

Step 5.3.3.10.1.3

Add $1$ and $2$ .

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

Step 5.3.3.10.2

Apply the distributive property.

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

Step 5.3.3.10.3

Multiply $2$ by $2$ .

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

Step 5.3.3.10.4

Multiply $- 1$ by $2$ .

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

Step 5.3.3.10.5

Apply the distributive property.

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

Step 5.3.3.10.6

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

Tap for more steps...

Step 5.3.3.10.6.1

Move $x$ .

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

Step 5.3.3.10.6.2

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

Tap for more steps...

Step 5.3.3.10.6.2.1

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

$y' = \frac{- y^{3} + 4 (x^{1} x^{2}) - 2 y^{2} x}{x y (2 x + 3 y)}$

Step 5.3.3.10.6.2.2

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

$y' = \frac{- y^{3} + 4 x^{1 + 2} - 2 y^{2} x}{x y (2 x + 3 y)}$

Step 5.3.3.10.6.3

Add $1$ and $2$ .

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

Step 5.3.3.11

Simplify with factoring out.

Tap for more steps...

Step 5.3.3.11.1

Factor $- 1$ out of $- y^{3}$ .

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

Step 5.3.3.11.2

Factor $- 1$ out of $4 x^{3}$ .

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

Step 5.3.3.11.3

Factor $- 1$ out of $- (y^{3}) - (- 4 x^{3})$ .

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

Step 5.3.3.11.4

Factor $- 1$ out of $- 2 y^{2} x$ .

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

Step 5.3.3.11.5

Factor $- 1$ out of $- (y^{3} - 4 x^{3}) - (2 y^{2} x)$ .

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

Step 5.3.3.11.6

Simplify the expression.

Tap for more steps...

Step 5.3.3.11.6.1

Rewrite $- (y^{3} - 4 x^{3} + 2 y^{2} x)$ as $- 1 (y^{3} - 4 x^{3} + 2 y^{2} x)$ .

$y' = \frac{- 1 (y^{3} - 4 x^{3} + 2 y^{2} x)}{x y (2 x + 3 y)}$

Step 5.3.3.11.6.2

Move the negative in front of the fraction.

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

Step 6

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

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