Calculus Examples

$x y^{2} - 2 x^{2} y = 1 + x^{2}$

Step 1

Differentiate both sides of the equation.

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

Step 2

Differentiate the left side of the equation.

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Step 2.1

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

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

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

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

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 $y$ .

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

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 $n = 2$ .

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

Step 2.2.2.3

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

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

Step 2.2.3

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

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

Step 2.2.4

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

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

Step 2.2.5

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

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

Step 2.2.6

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

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

Step 2.3

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

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Step 2.3.1

Since $- 2$ is constant with respect to $x$ , the derivative of $- 2 x^{2} y$ with respect to $x$ is $- 2 \frac{d}{d x} [x^{2} y]$ .

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

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

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

Step 2.3.3

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

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

Step 2.3.4

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

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

Step 2.3.5

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

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

Step 2.4

Simplify.

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Step 2.4.1

Apply the distributive property.

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

Step 2.4.2

Multiply $2$ by $- 2$ .

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

Step 2.4.3

Reorder terms.

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

Step 3

Differentiate the right side of the equation.

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Step 3.1

By the Sum Rule, the derivative of $1 + x^{2}$ with respect to $x$ is $\frac{d}{d x} [1] + \frac{d}{d x} [x^{2}]$ .

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

Step 3.2

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

$0 + \frac{d}{d x} [x^{2}]$

Step 3.3

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

$0 + 2 x$

Step 3.4

Add $0$ and $2 x$ .

$2 x$

Step 4

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

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

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.

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

Step 5.1.2

Add $4 x y$ to both sides of the equation.

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

Step 5.2

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

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Step 5.2.1

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

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

Step 5.2.2

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

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

Step 5.2.3

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

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

Step 5.3

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

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Step 5.3.1

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

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

Step 5.3.2

Simplify the left side.

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Step 5.3.2.1

Cancel the common factor of $2$ .

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Step 5.3.2.1.1

Cancel the common factor.

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

Step 5.3.2.1.2

Rewrite the expression.

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

Step 5.3.2.2

Cancel the common factor of $x$ .

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Step 5.3.2.2.1

Cancel the common factor.

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

Step 5.3.2.2.2

Rewrite the expression.

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

Step 5.3.2.3

Cancel the common factor of $y - x$ .

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Step 5.3.2.3.1

Cancel the common factor.

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

Step 5.3.2.3.2

Divide $y'$ by $1$ .

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

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

Cancel the common factor of $2$ .

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Step 5.3.3.1.1.1

Cancel the common factor.

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

Step 5.3.3.1.1.2

Rewrite the expression.

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

Step 5.3.3.1.2

Cancel the common factor of $x$ .

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Step 5.3.3.1.2.1

Cancel the common factor.

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

Step 5.3.3.1.2.2

Rewrite the expression.

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

Step 5.3.3.1.3

Move the negative in front of the fraction.

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

Step 5.3.3.1.4

Cancel the common factor of $4$ and $2$ .

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Step 5.3.3.1.4.1

Factor $2$ out of $4 x y$ .

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

Step 5.3.3.1.4.2

Cancel the common factors.

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Step 5.3.3.1.4.2.1

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

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

Step 5.3.3.1.4.2.2

Cancel the common factor.

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

Step 5.3.3.1.4.2.3

Rewrite the expression.

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

Step 5.3.3.1.5

Cancel the common factor of $x$ .

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Step 5.3.3.1.5.1

Cancel the common factor.

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

Step 5.3.3.1.5.2

Rewrite the expression.

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

Step 5.3.3.2

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

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

Step 5.3.3.3

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

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Step 5.3.3.3.1

Multiply $\frac{1}{y - x}$ by $\frac{2 x}{2 x}$ .

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

Step 5.3.3.3.2

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

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

Step 5.3.3.4

Combine the numerators over the common denominator.

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

Step 5.3.3.5

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

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

Step 5.3.3.6

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

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Step 5.3.3.6.1

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

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

Step 5.3.3.6.2

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

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

Step 5.3.3.7

Combine the numerators over the common denominator.

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

Step 5.3.3.8

Multiply $2$ by $2$ .

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

Step 6

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

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