Calculus Examples

$x^{2} y - \frac{x}{y} = - 2$

Step 1

Differentiate both sides of the equation.

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

Step 2

Differentiate the left side of the equation.

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Step 2.1

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

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

Step 2.2

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

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

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

Step 2.2.2

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

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

Step 2.2.3

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

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

Step 2.2.4

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

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

Step 2.3

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

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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 $f (x) = - 1$ and $g (x) = \frac{x}{y}$ .

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

Step 2.3.2

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 $g (x) = y$ .

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

Step 2.3.3

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

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

Step 2.3.4

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

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

Step 2.3.5

Since $- 1$ is constant with respect to $x$ , the derivative of $- 1$ with respect to $x$ is $0$ .

$x^{2} y' + 2 y x - \frac{y \cdot 1 - x y'}{y^{2}} + \frac{x}{y} \cdot 0$

Step 2.3.6

Multiply $y$ by $1$ .

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

Step 2.3.7

Multiply $\frac{x}{y}$ by $0$ .

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

Step 2.3.8

Add $- \frac{y - x y'}{y^{2}}$ and $0$ .

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

Step 2.4

Simplify.

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Step 2.4.1

Combine terms.

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Step 2.4.1.1

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

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

Step 2.4.1.2

Combine the numerators over the common denominator.

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

Step 2.4.2

Reorder terms.

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

Step 3

Since $- 2$ is constant with respect to $x$ , the derivative of $- 2$ with respect to $x$ is $0$ .

$0$

Step 4

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

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

Step 5

Solve for $y'$ .

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Step 5.1

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

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Step 5.1.1

Simplify the numerator.

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Step 5.1.1.1

Apply the distributive property.

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

Step 5.1.1.2

Multiply $- (- x y')$ .

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Step 5.1.1.2.1

Multiply $- 1$ by $- 1$ .

$2 x y + \frac{x^{2} y' y^{2} - y + 1 (x y')}{y^{2}} = 0$

Step 5.1.1.2.2

Multiply $x$ by $1$ .

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

Step 5.1.2

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

$\frac{2 x y \cdot y^{2}}{y^{2}} + \frac{x^{2} y' y^{2} - y + x y'}{y^{2}} = 0$

Step 5.1.3

Combine the numerators over the common denominator.

$\frac{2 x y \cdot y^{2} + x^{2} y' y^{2} - y + x y'}{y^{2}} = 0$

Step 5.1.4

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

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Step 5.1.4.1

Move $y^{2}$ .

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

Step 5.1.4.2

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

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Step 5.1.4.2.1

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

$\frac{2 x (y^{2} y^{1}) + x^{2} y' y^{2} - y + x y'}{y^{2}} = 0$

Step 5.1.4.2.2

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

$\frac{2 x y^{2 + 1} + x^{2} y' y^{2} - y + x y'}{y^{2}} = 0$

Step 5.1.4.3

Add $2$ and $1$ .

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

Step 5.2

Set the numerator equal to zero.

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

Step 5.3

Solve the equation for $y'$ .

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Step 5.3.1

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

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Step 5.3.1.1

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

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

Step 5.3.1.2

Add $y$ to both sides of the equation.

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

Step 5.3.2

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

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Step 5.3.2.1

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

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

Step 5.3.2.2

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

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

Step 5.3.2.3

Factor $x y'$ out of $x y' (x y^{2}) + x y' \cdot 1$ .

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

Step 5.3.3

Divide each term in $x y' (x y^{2} + 1) = - 2 x y^{3} + y$ by $x (x y^{2} + 1)$ and simplify.

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Step 5.3.3.1

Divide each term in $x y' (x y^{2} + 1) = - 2 x y^{3} + y$ by $x (x y^{2} + 1)$ .

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

Step 5.3.3.2

Simplify the left side.

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Step 5.3.3.2.1

Cancel the common factor of $x$ .

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Step 5.3.3.2.1.1

Cancel the common factor.

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

Step 5.3.3.2.1.2

Rewrite the expression.

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

Step 5.3.3.2.2

Cancel the common factor of $x y^{2} + 1$ .

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Step 5.3.3.2.2.1

Cancel the common factor.

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

Step 5.3.3.2.2.2

Divide $y'$ by $1$ .

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

Step 5.3.3.3

Simplify the right side.

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Step 5.3.3.3.1

Simplify each term.

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Step 5.3.3.3.1.1

Cancel the common factor of $x$ .

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Step 5.3.3.3.1.1.1

Cancel the common factor.

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

Step 5.3.3.3.1.1.2

Rewrite the expression.

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

Step 5.3.3.3.1.2

Move the negative in front of the fraction.

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

Step 5.3.3.3.2

To write $- \frac{2 y^{3}}{x y^{2} + 1}$ as a fraction with a common denominator, multiply by $\frac{x}{x}$ .

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

Step 5.3.3.3.3

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

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Step 5.3.3.3.3.1

Multiply $\frac{2 y^{3}}{x y^{2} + 1}$ by $\frac{x}{x}$ .

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

Step 5.3.3.3.3.2

Reorder the factors of $(x y^{2} + 1) x$ .

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

Step 5.3.3.3.4

Combine the numerators over the common denominator.

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

Step 5.3.3.3.5

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

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Step 5.3.3.3.5.1

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

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

Step 5.3.3.3.5.2

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

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

Step 5.3.3.3.5.3

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

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

Step 5.3.3.3.5.4

Factor $y$ out of $y (- 2 y^{2} x) + y \cdot 1$ .

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

Step 5.3.3.3.6

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

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

Step 5.3.3.3.7

Rewrite $1$ as $- 1 (- 1)$ .

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

Step 5.3.3.3.8

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

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

Step 5.3.3.3.9

Simplify the expression.

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Step 5.3.3.3.9.1

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

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

Step 5.3.3.3.9.2

Move the negative in front of the fraction.

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

Step 6

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

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