Calculus Examples

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

Step 1

Differentiate both sides of the equation.

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

Step 2

Differentiate the left side of the equation.

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Step 2.1

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

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

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 $\frac{d}{d x} [x] + \frac{d}{d x} [y]$ .

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

Step 2.2.2

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

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

Step 2.3

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

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

Step 2.4

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

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

Step 2.5

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

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

Step 2.6

Simplify.

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Step 2.6.1

Apply the distributive property.

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

Step 2.6.2

Apply the distributive property.

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

Step 2.6.3

Apply the distributive property.

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

Step 2.6.4

Combine terms.

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Step 2.6.4.1

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

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

Step 2.6.4.2

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

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Step 2.6.4.2.1

Move $x^{3}$ .

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

Step 2.6.4.2.2

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

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Step 2.6.4.2.2.1

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

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

Step 2.6.4.2.2.2

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

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

Step 2.6.4.2.3

Add $3$ and $1$ .

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

Step 2.6.4.3

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

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

Step 2.6.5

Reorder terms.

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

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 $g (x) = 3 x - y$ .

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

Step 3.2

Differentiate.

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Step 3.2.1

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

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

Step 3.2.2

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

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

Step 3.2.3

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

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

Step 3.2.4

Multiply $3$ by $1$ .

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

Step 3.2.5

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

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

Step 3.3

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

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

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 $y$ .

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

Step 3.4.2

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

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

Step 3.4.3

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

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

Step 3.5

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

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

Step 3.6

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

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

Step 3.7

Simplify.

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Step 3.7.1

Apply the distributive property.

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

Step 3.7.2

Apply the distributive property.

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

Step 3.7.3

Apply the distributive property.

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

Step 3.7.4

Apply the distributive property.

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

Step 3.7.5

Combine terms.

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Step 3.7.5.1

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

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

Step 3.7.5.2

Multiply $3$ by $2$ .

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

Step 3.7.5.3

Multiply $- 1$ by $2$ .

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

Step 3.7.5.4

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

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

Step 3.7.5.5

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

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

Step 3.7.5.6

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

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

Step 3.7.5.7

Add $1$ and $1$ .

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

Step 3.7.5.8

Subtract $2 y^{2} y'$ from $y^{2} (- y')$ .

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Step 3.7.5.8.1

Reorder $y^{2}$ and $- 1$ .

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

Step 3.7.5.8.2

Subtract $2 y^{2} y'$ from $- 1 \cdot y^{2} y'$ .

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

Step 4

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

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

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.

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

Step 5.2

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

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

Step 5.3

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

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

Step 5.4

Factor $y'$ out of $- 3 y^{2} y' + 6 x y y' - x^{4} y'$ .

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Step 5.4.1

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

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

Step 5.4.2

Factor $y'$ out of $6 x y y'$ .

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

Step 5.4.3

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

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

Step 5.4.4

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

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

Step 5.4.5

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

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

Step 5.5

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

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Step 5.5.1

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

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

Step 5.5.2

Simplify the left side.

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Step 5.5.2.1

Cancel the common factor of $- 3 y^{2} + 6 x y - x^{4}$ .

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Step 5.5.2.1.1

Cancel the common factor.

Step 5.5.2.1.2

Divide $y'$ by $1$ .

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

Step 5.5.3

Simplify the right side.

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Step 5.5.3.1

Move the negative in front of the fraction.

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

Step 5.5.3.2

Combine the numerators over the common denominator.

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

Step 5.5.3.3

Combine the numerators over the common denominator.

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

Step 5.5.3.4

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

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

Step 5.5.3.5

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

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

Step 5.5.3.6

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

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

Step 5.5.3.7

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

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

Step 5.5.3.8

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

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

Step 5.5.3.9

Rewrite negatives.

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Step 5.5.3.9.1

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

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

Step 5.5.3.9.2

Move the negative in front of the fraction.

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

Step 6

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

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