Calculus Examples

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

Step 1

Differentiate both sides of the equation.

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

Step 2

Differentiate the left side of the equation.

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

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Step 2.1.1

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

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

Step 2.1.2

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

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

Step 2.1.3

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

$2 y \frac{d}{d x} [y]$

$2 y \frac{d}{d x} [y]$

Step 2.2

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

$2 y y'$

$2 y y'$

Step 3

Differentiate the right side of the equation.

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

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

Step 3.2

Differentiate.

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Step 3.2.1

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

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

Step 3.2.2

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

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

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

Step 3.3

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

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

Step 3.4

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

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

Step 3.5

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

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

Step 3.6

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

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

Step 3.7

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

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

Step 3.8

Simplify.

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Step 3.8.1

Apply the distributive property.

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

Step 3.8.2

Apply the distributive property.

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

Step 3.8.3

Apply the distributive property.

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

Step 3.8.4

Apply the distributive property.

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

Step 3.8.5

Apply the distributive property.

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

Step 3.8.6

Apply the distributive property.

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

Step 3.8.7

Combine terms.

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Step 3.8.7.1

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

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

Step 3.8.7.2

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

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

Step 3.8.7.3

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

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

Step 3.8.7.4

Add $1$ and $1$ .

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

Step 3.8.7.5

Multiply $2$ by $- 1$ .

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

Step 3.8.7.6

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

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

Step 3.8.7.7

Multiply $y$ by $1$ .

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

Step 3.8.7.8

Add $2 x^{2}$ and $x^{2}$ .

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

Step 3.8.7.9

Add $- y y'$ and $y (- y')$ .

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Step 3.8.7.9.1

Reorder $y$ and $- 1$ .

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

Step 3.8.7.9.2

Subtract $y y'$ from $- y y'$ .

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

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

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

Step 3.8.8

Reorder terms.

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

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

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

Step 4

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

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

Step 5

Solve for $y'$ .

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Step 5.1

Since $y'$ is on the right side of the equation, switch the sides so it is on the left side of the equation.

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

Step 5.2

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

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Step 5.2.1

Subtract $2 y y'$ from both sides of the equation.

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

Step 5.2.2

Subtract $2 y y'$ from $- 2 y y'$ .

$3 x^{2} - 2 x y + x y' - 4 y y' - x^{2} y' + y = 0$

$3 x^{2} - 2 x y + x y' - 4 y y' - x^{2} y' + y = 0$

Step 5.3

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

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Step 5.3.1

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

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

Step 5.3.2

Add $2 x y$ to both sides of the equation.

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

Step 5.3.3

Subtract $y$ from both sides of the equation.

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

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

Step 5.4

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

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Step 5.4.1

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

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

Step 5.4.2

Factor $y'$ out of $- 4 y y'$ .

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

Step 5.4.3

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

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

Step 5.4.4

Factor $y'$ out of $y' (x) + y' (- 4 y)$ .

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

Step 5.4.5

Factor $y'$ out of $y' (x - 4 y) + y' (- x^{2})$ .

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

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

Step 5.5

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

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Step 5.5.1

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

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

Step 5.5.2

Simplify the left side.

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Step 5.5.2.1

Cancel the common factor of $x - 4 y - x^{2}$ .

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Step 5.5.2.1.1

Cancel the common factor.

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

Step 5.5.2.1.2

Divide $y'$ by $1$ .

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

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

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

Step 5.5.3

Simplify the right side.

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Step 5.5.3.1

Simplify each term.

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Step 5.5.3.1.1

Move the negative in front of the fraction.

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

Step 5.5.3.1.2

Move the negative in front of the fraction.

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

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

Step 5.5.3.2

Simplify terms.

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Step 5.5.3.2.1

Combine the numerators over the common denominator.

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

Step 5.5.3.2.2

Combine the numerators over the common denominator.

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

Step 5.5.3.2.3

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

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

Step 5.5.3.2.4

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

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

Step 5.5.3.2.5

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

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

Step 5.5.3.2.6

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

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

Step 5.5.3.2.7

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

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

Step 5.5.3.2.8

Simplify the expression.

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Step 5.5.3.2.8.1

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

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

Step 5.5.3.2.8.2

Move the negative in front of the fraction.

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

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

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

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

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

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

Step 6

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

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

$[\begin{array}{c} x^{2} & \frac{1}{2} \\ \sqrt{π} & \int x d x \end{array}]$