Calculus Examples

$3 x^{2} - 4 y^{3} = 12 x^{3} y$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (3 x^{2} - 4 y^{3}) = \frac{d}{d x} (12 x^{3} y)$

Step 2

Differentiate the left side of the equation.

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Step 2.1

By the Sum Rule, the derivative of $3 x^{2} - 4 y^{3}$ with respect to $x$ is $\frac{d}{d x} [3 x^{2}] + \frac{d}{d x} [- 4 y^{3}]$ .

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

Step 2.2

Evaluate $\frac{d}{d x} [3 x^{2}]$ .

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Step 2.2.1

Since $3$ is constant with respect to $x$ , the derivative of $3 x^{2}$ with respect to $x$ is $3 \frac{d}{d x} [x^{2}]$ .

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

Step 2.2.2

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

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

Step 2.2.3

Multiply $2$ by $3$ .

$6 x + \frac{d}{d x} [- 4 y^{3}]$

Step 2.3

Evaluate $\frac{d}{d x} [- 4 y^{3}]$ .

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Step 2.3.1

Since $- 4$ is constant with respect to $x$ , the derivative of $- 4 y^{3}$ with respect to $x$ is $- 4 \frac{d}{d x} [y^{3}]$ .

$6 x - 4 \frac{d}{d x} [y^{3}]$

Step 2.3.2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{3}$ and $g (x) = y$ .

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Step 2.3.2.1

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

$6 x - 4 (\frac{d}{d u} [u^{3}] \frac{d}{d x} [y])$

Step 2.3.2.2

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = 3$ .

$6 x - 4 (3 u^{2} \frac{d}{d x} [y])$

Step 2.3.2.3

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

$6 x - 4 (3 y^{2} \frac{d}{d x} [y])$

Step 2.3.3

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

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

Step 2.3.4

Multiply $3$ by $- 4$ .

$6 x - 12 y^{2} y'$

Step 2.4

Reorder terms.

$- 12 y^{2} y' + 6 x$

Step 3

Differentiate the right side of the equation.

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Step 3.1

Since $12$ is constant with respect to $x$ , the derivative of $12 x^{3} y$ with respect to $x$ is $12 \frac{d}{d x} [x^{3} y]$ .

$12 \frac{d}{d x} [x^{3} y]$

Step 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 $f (x) = x^{3}$ and $g (x) = y$ .

$12 (x^{3} \frac{d}{d x} [y] + y \frac{d}{d x} [x^{3}])$

Step 3.3

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

$12 (x^{3} y' + y \frac{d}{d x} [x^{3}])$

Step 3.4

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 3$ .

$12 (x^{3} y' + y (3 x^{2}))$

Step 3.5

Move $3$ to the left of $y$ .

$12 (x^{3} y' + 3 \cdot y x^{2})$

Step 3.6

Simplify.

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Step 3.6.1

Apply the distributive property.

$12 (x^{3} y') + 12 (3 y x^{2})$

Step 3.6.2

Multiply $3$ by $12$ .

$12 x^{3} y' + 36 y x^{2}$

Step 3.6.3

Reorder terms.

$12 x^{3} y' + 36 x^{2} y$

Step 4

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

$- 12 y^{2} y' + 6 x = 12 x^{3} y' + 36 x^{2} y$

Step 5

Solve for $y'$ .

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Step 5.1

Subtract $12 x^{3} y'$ from both sides of the equation.

$- 12 y^{2} y' + 6 x - 12 x^{3} y' = 36 x^{2} y$

Step 5.2

Subtract $6 x$ from both sides of the equation.

$- 12 y^{2} y' - 12 x^{3} y' = 36 x^{2} y - 6 x$

Step 5.3

Factor $12 y'$ out of $- 12 y^{2} y' - 12 x^{3} y'$ .

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Step 5.3.1

Factor $12 y'$ out of $- 12 y^{2} y'$ .

$12 y' (- y^{2}) - 12 x^{3} y' = 36 x^{2} y - 6 x$

Step 5.3.2

Factor $12 y'$ out of $- 12 x^{3} y'$ .

$12 y' (- y^{2}) + 12 y' (- x^{3}) = 36 x^{2} y - 6 x$

Step 5.3.3

Factor $12 y'$ out of $12 y' (- y^{2}) + 12 y' (- x^{3})$ .

$12 y' (- y^{2} - x^{3}) = 36 x^{2} y - 6 x$

Step 5.4

Divide each term in $12 y' (- y^{2} - x^{3}) = 36 x^{2} y - 6 x$ by $12 (- y^{2} - x^{3})$ and simplify.

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Step 5.4.1

Divide each term in $12 y' (- y^{2} - x^{3}) = 36 x^{2} y - 6 x$ by $12 (- y^{2} - x^{3})$ .

$\frac{12 y' (- y^{2} - x^{3})}{12 (- y^{2} - x^{3})} = \frac{36 x^{2} y}{12 (- y^{2} - x^{3})} + \frac{- 6 x}{12 (- y^{2} - x^{3})}$

Step 5.4.2

Simplify the left side.

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Step 5.4.2.1

Cancel the common factor of $12$ .

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Step 5.4.2.1.1

Cancel the common factor.

$\frac{12 y' (- y^{2} - x^{3})}{12 (- y^{2} - x^{3})} = \frac{36 x^{2} y}{12 (- y^{2} - x^{3})} + \frac{- 6 x}{12 (- y^{2} - x^{3})}$

Step 5.4.2.1.2

Rewrite the expression.

$\frac{y' (- y^{2} - x^{3})}{- y^{2} - x^{3}} = \frac{36 x^{2} y}{12 (- y^{2} - x^{3})} + \frac{- 6 x}{12 (- y^{2} - x^{3})}$

Step 5.4.2.2

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

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Step 5.4.2.2.1

Cancel the common factor.

$\frac{y' (- y^{2} - x^{3})}{- y^{2} - x^{3}} = \frac{36 x^{2} y}{12 (- y^{2} - x^{3})} + \frac{- 6 x}{12 (- y^{2} - x^{3})}$

Step 5.4.2.2.2

Divide $y'$ by $1$ .

$y' = \frac{36 x^{2} y}{12 (- y^{2} - x^{3})} + \frac{- 6 x}{12 (- y^{2} - x^{3})}$

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

Cancel the common factor of $36$ and $12$ .

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Step 5.4.3.1.1.1

Factor $12$ out of $36 x^{2} y$ .

$y' = \frac{12 (3 x^{2} y)}{12 (- y^{2} - x^{3})} + \frac{- 6 x}{12 (- y^{2} - x^{3})}$

Step 5.4.3.1.1.2

Cancel the common factors.

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Step 5.4.3.1.1.2.1

Cancel the common factor.

$y' = \frac{12 (3 x^{2} y)}{12 (- y^{2} - x^{3})} + \frac{- 6 x}{12 (- y^{2} - x^{3})}$

Step 5.4.3.1.1.2.2

Rewrite the expression.

$y' = \frac{3 x^{2} y}{- y^{2} - x^{3}} + \frac{- 6 x}{12 (- y^{2} - x^{3})}$

Step 5.4.3.1.2

Cancel the common factor of $- 6$ and $12$ .

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Step 5.4.3.1.2.1

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

$y' = \frac{3 x^{2} y}{- y^{2} - x^{3}} + \frac{6 (- x)}{12 (- y^{2} - x^{3})}$

Step 5.4.3.1.2.2

Cancel the common factors.

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Step 5.4.3.1.2.2.1

Factor $6$ out of $12 (- y^{2} - x^{3})$ .

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

Step 5.4.3.1.2.2.2

Cancel the common factor.

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

Step 5.4.3.1.2.2.3

Rewrite the expression.

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

Step 5.4.3.1.3

Move the negative in front of the fraction.

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

Step 5.4.3.2

To write $\frac{3 x^{2} y}{- y^{2} - x^{3}}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Step 5.4.3.3

Write each expression with a common denominator of $(- y^{2} - x^{3}) \cdot 2$ , by multiplying each by an appropriate factor of $1$ .

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Step 5.4.3.3.1

Multiply $\frac{3 x^{2} y}{- y^{2} - x^{3}}$ by $\frac{2}{2}$ .

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

Step 5.4.3.3.2

Reorder the factors of $(- y^{2} - x^{3}) \cdot 2$ .

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

Step 5.4.3.4

Combine the numerators over the common denominator.

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

Step 5.4.3.5

Simplify the numerator.

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Step 5.4.3.5.1

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

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Step 5.4.3.5.1.1

Factor $x$ out of $3 x^{2} y \cdot 2$ .

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

Step 5.4.3.5.1.2

Factor $x$ out of $- x$ .

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

Step 5.4.3.5.1.3

Factor $x$ out of $x (3 x y \cdot 2) + x \cdot - 1$ .

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

Step 5.4.3.5.2

Multiply $2$ by $3$ .

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

Step 5.4.3.6

Simplify with factoring out.

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Step 5.4.3.6.1

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

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

Step 5.4.3.6.2

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

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

Step 5.4.3.6.3

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

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

Step 5.4.3.6.4

Rewrite negatives.

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Step 5.4.3.6.4.1

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

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

Step 5.4.3.6.4.2

Move the negative in front of the fraction.

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

Step 6

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

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