Calculus Examples

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

Step 1

Differentiate both sides of the equation.

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

Step 2

Differentiate the left side of the equation.

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Step 2.1

Differentiate using the Product Rule which states that $\frac{d}{d y} [f (y) g (y)]$ is $f (y) \frac{d}{d y} [g (y)] + g (y) \frac{d}{d y} [f (y)]$ where $f (y) = x^{4}$ and $g (y) = x + y$ .

$x^{4} \frac{d}{d y} [x + y] + (x + y) \frac{d}{d y} [x^{4}]$

Step 2.2

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

$x^{4} (\frac{d}{d y} [x] + \frac{d}{d y} [y]) + (x + y) \frac{d}{d y} [x^{4}]$

Step 2.3

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

$x^{4} (x' + \frac{d}{d y} [y]) + (x + y) \frac{d}{d y} [x^{4}]$

Step 2.4

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

$x^{4} (x' + 1) + (x + y) \frac{d}{d y} [x^{4}]$

Step 2.5

Differentiate using the chain rule, which states that $\frac{d}{d y} [f (g (y))]$ is $f' (g (y)) g' (y)$ where $f (y) = y^{4}$ and $g (y) = x$ .

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Step 2.5.1

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

$x^{4} (x' + 1) + (x + y) (\frac{d}{d u} [u^{4}] \frac{d}{d y} [x])$

Step 2.5.2

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = 4$ .

$x^{4} (x' + 1) + (x + y) (4 u^{3} \frac{d}{d y} [x])$

Step 2.5.3

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

$x^{4} (x' + 1) + (x + y) (4 x^{3} \frac{d}{d y} [x])$

Step 2.6

Move $4$ to the left of $x + y$ .

$x^{4} (x' + 1) + 4 \cdot (x + y) x^{3} \frac{d}{d y} [x]$

Step 2.7

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

$x^{4} (x' + 1) + 4 (x + y) x^{3} x'$

Step 2.8

Simplify.

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Step 2.8.1

Apply the distributive property.

$x^{4} x' + x^{4} \cdot 1 + 4 (x + y) x^{3} x'$

Step 2.8.2

Apply the distributive property.

$x^{4} x' + x^{4} \cdot 1 + (4 x + 4 y) x^{3} x'$

Step 2.8.3

Apply the distributive property.

$x^{4} x' + x^{4} \cdot 1 + (4 x \cdot x^{3} + 4 y x^{3}) x'$

Step 2.8.4

Apply the distributive property.

$x^{4} x' + x^{4} \cdot 1 + 4 x \cdot x^{3} x' + 4 y x^{3} x'$

Step 2.8.5

Combine terms.

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Step 2.8.5.1

Multiply $x^{4}$ by $1$ .

$x^{4} x' + x^{4} + 4 x \cdot x^{3} x' + 4 y x^{3} x'$

Step 2.8.5.2

Multiply $x$ by $x^{3}$ by adding the exponents.

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Step 2.8.5.2.1

Move $x^{3}$ .

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

Step 2.8.5.2.2

Multiply $x^{3}$ by $x$ .

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Step 2.8.5.2.2.1

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

$x^{4} x' + x^{4} + 4 (x^{3} x^{1}) x' + 4 y x^{3} x'$

Step 2.8.5.2.2.2

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

$x^{4} x' + x^{4} + 4 x^{3 + 1} x' + 4 y x^{3} x'$

Step 2.8.5.2.3

Add $3$ and $1$ .

$x^{4} x' + x^{4} + 4 x^{4} x' + 4 y x^{3} x'$

Step 2.8.5.3

Add $x^{4} x'$ and $4 x^{4} x'$ .

$x^{4} + 5 x^{4} x' + 4 y x^{3} x'$

Step 2.8.6

Reorder terms.

$x^{4} + 5 x^{4} x' + 4 x^{3} y x'$

Step 3

Differentiate the right side of the equation.

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Step 3.1

Differentiate using the Product Rule which states that $\frac{d}{d y} [f (y) g (y)]$ is $f (y) \frac{d}{d y} [g (y)] + g (y) \frac{d}{d y} [f (y)]$ where $f (y) = y^{2}$ and $g (y) = 3 x - y$ .

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

Step 3.2

Differentiate.

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Step 3.2.1

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

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

Step 3.2.2

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

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

Step 3.3

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

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

Step 3.4

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

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

Step 3.5

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

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

Step 3.6

Multiply $- 1$ by $1$ .

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

Step 3.7

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

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

Step 3.8

Move $2$ to the left of $3 x - y$ .

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

Step 3.9

Simplify.

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Step 3.9.1

Apply the distributive property.

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

Step 3.9.2

Apply the distributive property.

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

Step 3.9.3

Apply the distributive property.

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

Step 3.9.4

Combine terms.

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Step 3.9.4.1

Move $3$ to the left of $y^{2}$ .

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

Step 3.9.4.2

Move $- 1$ to the left of $y^{2}$ .

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

Step 3.9.4.3

Rewrite $- 1 y^{2}$ as $- y^{2}$ .

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

Step 3.9.4.4

Multiply $3$ by $2$ .

$3 y^{2} x' - y^{2} + 6 x y + 2 (- y) y$

Step 3.9.4.5

Multiply $- 1$ by $2$ .

$3 y^{2} x' - y^{2} + 6 x y - 2 y \cdot y$

Step 3.9.4.6

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

$3 y^{2} x' - y^{2} + 6 x y - 2 (y^{1} y)$

Step 3.9.4.7

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

$3 y^{2} x' - y^{2} + 6 x y - 2 (y^{1} y^{1})$

Step 3.9.4.8

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

$3 y^{2} x' - y^{2} + 6 x y - 2 y^{1 + 1}$

Step 3.9.4.9

Add $1$ and $1$ .

$3 y^{2} x' - y^{2} + 6 x y - 2 y^{2}$

Step 3.9.4.10

Subtract $2 y^{2}$ from $- y^{2}$ .

$3 y^{2} x' - 3 y^{2} + 6 x y$

Step 3.9.5

Reorder terms.

$- 3 y^{2} + 3 y^{2} x' + 6 x y$

Step 4

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

$x^{4} + 5 x^{4} x' + 4 x^{3} y x' = - 3 y^{2} + 3 y^{2} x' + 6 x y$

Step 5

Solve for $x'$ .

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Step 5.1

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

$x^{4} + 5 x^{4} x' + 4 x^{3} y x' - 3 y^{2} x' = - 3 y^{2} + 6 x y$

Step 5.2

Subtract $x^{4}$ from both sides of the equation.

$5 x^{4} x' + 4 x^{3} y x' - 3 y^{2} x' = - 3 y^{2} + 6 x y - x^{4}$

Step 5.3

Factor $x'$ out of $5 x^{4} x' + 4 x^{3} y x' - 3 y^{2} x'$ .

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Step 5.3.1

Factor $x'$ out of $5 x^{4} x'$ .

$x' (5 x^{4}) + 4 x^{3} y x' - 3 y^{2} x' = - 3 y^{2} + 6 x y - x^{4}$

Step 5.3.2

Factor $x'$ out of $4 x^{3} y x'$ .

$x' (5 x^{4}) + x' (4 x^{3} y) - 3 y^{2} x' = - 3 y^{2} + 6 x y - x^{4}$

Step 5.3.3

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

$x' (5 x^{4}) + x' (4 x^{3} y) + x' (- 3 y^{2}) = - 3 y^{2} + 6 x y - x^{4}$

Step 5.3.4

Factor $x'$ out of $x' (5 x^{4}) + x' (4 x^{3} y)$ .

$x' (5 x^{4} + 4 x^{3} y) + x' (- 3 y^{2}) = - 3 y^{2} + 6 x y - x^{4}$

Step 5.3.5

Factor $x'$ out of $x' (5 x^{4} + 4 x^{3} y) + x' (- 3 y^{2})$ .

$x' (5 x^{4} + 4 x^{3} y - 3 y^{2}) = - 3 y^{2} + 6 x y - x^{4}$

Step 5.4

Divide each term in $x' (5 x^{4} + 4 x^{3} y - 3 y^{2}) = - 3 y^{2} + 6 x y - x^{4}$ by $5 x^{4} + 4 x^{3} y - 3 y^{2}$ and simplify.

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Step 5.4.1

Divide each term in $x' (5 x^{4} + 4 x^{3} y - 3 y^{2}) = - 3 y^{2} + 6 x y - x^{4}$ by $5 x^{4} + 4 x^{3} y - 3 y^{2}$ .

$\frac{x' (5 x^{4} + 4 x^{3} y - 3 y^{2})}{5 x^{4} + 4 x^{3} y - 3 y^{2}} = \frac{- 3 y^{2}}{5 x^{4} + 4 x^{3} y - 3 y^{2}} + \frac{6 x y}{5 x^{4} + 4 x^{3} y - 3 y^{2}} + \frac{- x^{4}}{5 x^{4} + 4 x^{3} y - 3 y^{2}}$

Step 5.4.2

Simplify the left side.

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Step 5.4.2.1

Cancel the common factor of $5 x^{4} + 4 x^{3} y - 3 y^{2}$ .

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Step 5.4.2.1.1

Cancel the common factor.

Step 5.4.2.1.2

Divide $x'$ by $1$ .

$x' = \frac{- 3 y^{2}}{5 x^{4} + 4 x^{3} y - 3 y^{2}} + \frac{6 x y}{5 x^{4} + 4 x^{3} y - 3 y^{2}} + \frac{- x^{4}}{5 x^{4} + 4 x^{3} y - 3 y^{2}}$

Step 5.4.3

Simplify the right side.

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Step 5.4.3.1

Simplify each term.

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Step 5.4.3.1.1

Move the negative in front of the fraction.

$x' = - \frac{(3) y^{2}}{5 x^{4} + 4 x^{3} y - 3 y^{2}} + \frac{6 x y}{5 x^{4} + 4 x^{3} y - 3 y^{2}} + \frac{- x^{4}}{5 x^{4} + 4 x^{3} y - 3 y^{2}}$

Step 5.4.3.1.2

Move the negative in front of the fraction.

$x' = - \frac{3 y^{2}}{5 x^{4} + 4 x^{3} y - 3 y^{2}} + \frac{6 x y}{5 x^{4} + 4 x^{3} y - 3 y^{2}} - \frac{x^{4}}{5 x^{4} + 4 x^{3} y - 3 y^{2}}$

Step 5.4.3.2

Simplify terms.

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Step 5.4.3.2.1

Combine the numerators over the common denominator.

$x' = \frac{- 3 y^{2} + 6 x y}{5 x^{4} + 4 x^{3} y - 3 y^{2}} - \frac{x^{4}}{5 x^{4} + 4 x^{3} y - 3 y^{2}}$

Step 5.4.3.2.2

Combine the numerators over the common denominator.

$x' = \frac{- 3 y^{2} + 6 x y - x^{4}}{5 x^{4} + 4 x^{3} y - 3 y^{2}}$

Step 5.4.3.2.3

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

$x' = \frac{- (3 y^{2}) + 6 x y - x^{4}}{5 x^{4} + 4 x^{3} y - 3 y^{2}}$

Step 5.4.3.2.4

Factor $- 1$ out of $6 x y$ .

$x' = \frac{- (3 y^{2}) - (- 6 x y) - x^{4}}{5 x^{4} + 4 x^{3} y - 3 y^{2}}$

Step 5.4.3.2.5

Factor $- 1$ out of $- (3 y^{2}) - (- 6 x y)$ .

$x' = \frac{- (3 y^{2} - 6 x y) - x^{4}}{5 x^{4} + 4 x^{3} y - 3 y^{2}}$

Step 5.4.3.2.6

Factor $- 1$ out of $- x^{4}$ .

$x' = \frac{- (3 y^{2} - 6 x y) - (x^{4})}{5 x^{4} + 4 x^{3} y - 3 y^{2}}$

Step 5.4.3.2.7

Factor $- 1$ out of $- (3 y^{2} - 6 x y) - (x^{4})$ .

$x' = \frac{- (3 y^{2} - 6 x y + x^{4})}{5 x^{4} + 4 x^{3} y - 3 y^{2}}$

Step 5.4.3.2.8

Simplify the expression.

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Step 5.4.3.2.8.1

Rewrite $- (3 y^{2} - 6 x y + x^{4})$ as $- 1 (3 y^{2} - 6 x y + x^{4})$ .

$x' = \frac{- 1 (3 y^{2} - 6 x y + x^{4})}{5 x^{4} + 4 x^{3} y - 3 y^{2}}$

Step 5.4.3.2.8.2

Move the negative in front of the fraction.

$x' = - \frac{3 y^{2} - 6 x y + x^{4}}{5 x^{4} + 4 x^{3} y - 3 y^{2}}$

Step 6

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

$\frac{d x}{d y} = - \frac{3 y^{2} - 6 x y + x^{4}}{5 x^{4} + 4 x^{3} y - 3 y^{2}}$