Calculus Examples

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

Step 1

Differentiate both sides of the equation.

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

Step 2

Differentiate the left side of the equation.

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Step 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 $g (x) = x - y$ .

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

Step 2.2

Differentiate.

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Step 2.2.1

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

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

Step 2.2.2

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

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

Step 2.2.3

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

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

Step 2.3

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

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

Step 2.4

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

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

Step 2.5

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

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

Step 2.6

Simplify.

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Step 2.6.1

Apply the distributive property.

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

Step 2.6.2

Apply the distributive property.

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

Step 2.6.3

Apply the distributive property.

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

Step 2.6.4

Combine terms.

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Step 2.6.4.1

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

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

Step 2.6.4.2

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

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

Step 2.6.4.3

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

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

Step 2.6.4.4

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

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

Step 2.6.4.5

Add $1$ and $1$ .

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

Step 2.6.4.6

Multiply $- 1$ by $2$ .

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

Step 2.6.4.7

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

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

Step 2.6.5

Reorder terms.

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

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

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

Step 3.2

Differentiate.

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Step 3.2.1

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

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

Step 3.2.2

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

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

Step 3.3

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

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

Step 3.4

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 3.4.1

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

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

Step 3.4.2

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

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

Step 3.4.3

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

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

Step 3.5

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

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

Step 3.6

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

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

Step 3.7

Simplify.

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Step 3.7.1

Apply the distributive property.

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

Step 3.7.2

Apply the distributive property.

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

Step 3.7.3

Apply the distributive property.

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

Step 3.7.4

Apply the distributive property.

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

Step 3.7.5

Combine terms.

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Step 3.7.5.1

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

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

Step 3.7.5.2

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

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

Step 3.7.5.3

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

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

Step 3.7.5.4

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

$y^{2} + y^{2} y' + 2 x y y' + 2 y^{1 + 1} y'$

Step 3.7.5.5

Add $1$ and $1$ .

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

Step 3.7.5.6

Add $y^{2} y'$ and $2 y^{2} y'$ .

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

Step 4

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

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

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.

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

Step 5.2

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

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

Step 5.3

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

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

Step 5.4

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

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Step 5.4.1

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

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

Step 5.4.2

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

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

Step 5.4.3

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

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

Step 5.4.4

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

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

Step 5.4.5

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

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

Step 5.5

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

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Step 5.5.1

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

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

Step 5.5.2

Simplify the left side.

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Step 5.5.2.1

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

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Step 5.5.2.1.1

Cancel the common factor.

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

Step 5.5.2.1.2

Divide $y'$ by $1$ .

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

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.

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

Step 5.5.3.1.2

Move the negative in front of the fraction.

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

Step 5.5.3.2

Combine into one fraction.

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Step 5.5.3.2.1

Combine the numerators over the common denominator.

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

Step 5.5.3.2.2

Combine the numerators over the common denominator.

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

Step 5.5.3.3

Factor by grouping.

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Step 5.5.3.3.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 3 \cdot - 1 = - 3$ and whose sum is $b = - 2$ .

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Step 5.5.3.3.1.1

Reorder terms.

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

Step 5.5.3.3.1.2

Reorder $- y^{2}$ and $- 2 x y$ .

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

Step 5.5.3.3.1.3

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

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

Step 5.5.3.3.1.4

Rewrite $- 2$ as $1$ plus $- 3$

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

Step 5.5.3.3.1.5

Apply the distributive property.

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

Step 5.5.3.3.1.6

Multiply $x y$ by $1$ .

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

Step 5.5.3.3.1.7

Move parentheses.

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

Step 5.5.3.3.2

Factor out the greatest common factor from each group.

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Step 5.5.3.3.2.1

Group the first two terms and the last two terms.

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

Step 5.5.3.3.2.2

Factor out the greatest common factor (GCF) from each group.

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

Step 5.5.3.3.3

Factor the polynomial by factoring out the greatest common factor, $3 x + y$ .

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

Step 6

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

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