Calculus Examples

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

Step 1

Differentiate both sides of the equation.

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

Step 2

Differentiate the left side of the equation.

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Step 2.1

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

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

Step 2.2

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

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

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

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 $g (x) = y$ .

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Step 2.2.2.1

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

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

Step 2.2.2.2

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

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

Step 2.2.2.3

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

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

Step 2.2.3

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

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

Step 2.2.4

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

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

Step 2.2.5

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

$2 \cdot x y y' + y^{2} \cdot 1 + \frac{d}{d x} [- x^{3} y]$

Step 2.2.6

Multiply $y^{2}$ by $1$ .

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

Step 2.3

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

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Step 2.3.1

Since $- 1$ is constant with respect to $x$ , the derivative of $- x^{3} y$ with respect to $x$ is $- \frac{d}{d x} [x^{3} y]$ .

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

Step 2.3.2

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

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

Step 2.3.3

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

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

Step 2.3.4

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

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

Step 2.3.5

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

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

Step 2.4

Simplify.

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Step 2.4.1

Apply the distributive property.

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

Step 2.4.2

Multiply $3$ by $- 1$ .

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

Step 2.4.3

Reorder terms.

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

Step 3

Since $6$ is constant with respect to $x$ , the derivative of $6$ with respect to $x$ is $0$ .

$0$

Step 4

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

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

Step 5

Solve for $y'$ .

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Step 5.1

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

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Step 5.1.1

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

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

Step 5.1.2

Add $3 x^{2} y$ to both sides of the equation.

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

Step 5.2

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

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Step 5.2.1

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

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

Step 5.2.2

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

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

Step 5.2.3

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

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

Step 5.3

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

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Step 5.3.1

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

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

Step 5.3.2

Simplify the left side.

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Step 5.3.2.1

Cancel the common factor of $x$ .

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Step 5.3.2.1.1

Cancel the common factor.

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

Step 5.3.2.1.2

Rewrite the expression.

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

Step 5.3.2.2

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

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Step 5.3.2.2.1

Cancel the common factor.

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

Step 5.3.2.2.2

Divide $y'$ by $1$ .

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

Step 5.3.3

Simplify the right side.

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Step 5.3.3.1

Simplify each term.

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Step 5.3.3.1.1

Move the negative in front of the fraction.

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

Step 5.3.3.1.2

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

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Step 5.3.3.1.2.1

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

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

Step 5.3.3.1.2.2

Cancel the common factors.

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Step 5.3.3.1.2.2.1

Cancel the common factor.

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

Step 5.3.3.1.2.2.2

Rewrite the expression.

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

Step 5.3.3.2

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

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

Step 5.3.3.3

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

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Step 5.3.3.3.1

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

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

Step 5.3.3.3.2

Reorder the factors of $(2 y - x^{2}) x$ .

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

Step 5.3.3.4

Combine the numerators over the common denominator.

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

Step 5.3.3.5

Simplify the numerator.

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Step 5.3.3.5.1

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

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Step 5.3.3.5.1.1

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

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

Step 5.3.3.5.1.2

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

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

Step 5.3.3.5.1.3

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

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

Step 5.3.3.5.2

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

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Step 5.3.3.5.2.1

Move $x$ .

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

Step 5.3.3.5.2.2

Multiply $x$ by $x$ .

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

Step 5.3.3.6

Simplify with factoring out.

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Step 5.3.3.6.1

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

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

Step 5.3.3.6.2

Factor $- 1$ out of $3 x^{2}$ .

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

Step 5.3.3.6.3

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

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

Step 5.3.3.6.4

Simplify the expression.

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Step 5.3.3.6.4.1

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

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

Step 5.3.3.6.4.2

Move the negative in front of the fraction.

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

Step 6

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

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