Calculus Examples

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

Step 1

Differentiate both sides of the equation.

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

Step 2

Differentiate the left side of the equation.

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

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 2.3.1

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

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

Step 2.3.2

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

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

Step 2.3.3

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

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

Step 2.4

Move $2$ to the left of $x^{2}$ .

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

Step 2.5

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

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

Step 2.6

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

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

Step 2.7

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

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

Step 2.8

Simplify.

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Step 2.8.1

Apply the distributive property.

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

Step 2.8.2

Combine terms.

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Step 2.8.2.1

Multiply $2$ by $3$ .

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

Step 2.8.2.2

Multiply $2$ by $3$ .

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

Step 3

Differentiate the right side of the equation.

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Step 3.1

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

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

Step 3.2

Evaluate $\frac{d}{d x} [4 x^{2}]$ .

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Step 3.2.1

Since $4$ is constant with respect to $x$ , the derivative of $4 x^{2}$ with respect to $x$ is $4 \frac{d}{d x} [x^{2}]$ .

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

Step 3.2.2

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

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

Step 3.2.3

Multiply $2$ by $4$ .

$8 x + \frac{d}{d x} [- 4 x y]$

Step 3.3

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

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Step 3.3.1

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

$8 x - 4 \frac{d}{d x} [x y]$

Step 3.3.2

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 $f (x) = x$ and $g (x) = y$ .

$8 x - 4 (x \frac{d}{d x} [y] + y \frac{d}{d x} [x])$

Step 3.3.3

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

$8 x - 4 (x y' + y \frac{d}{d x} [x])$

Step 3.3.4

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

$8 x - 4 (x y' + y \cdot 1)$

Step 3.3.5

Multiply $y$ by $1$ .

$8 x - 4 (x y' + y)$

Step 3.4

Simplify.

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Step 3.4.1

Apply the distributive property.

$8 x - 4 (x y') - 4 y$

Step 3.4.2

Remove unnecessary parentheses.

$8 x - 4 x y' - 4 y$

Step 3.4.3

Reorder terms.

$- 4 x y' + 8 x - 4 y$

Step 4

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

$6 x^{2} y y' + 6 y^{2} x = - 4 x y' + 8 x - 4 y$

Step 5

Solve for $y'$ .

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Step 5.1

Add $4 x y'$ to both sides of the equation.

$6 x^{2} y y' + 6 y^{2} x + 4 x y' = 8 x - 4 y$

Step 5.2

Subtract $6 y^{2} x$ from both sides of the equation.

$6 x^{2} y y' + 4 x y' = 8 x - 4 y - 6 y^{2} x$

Step 5.3

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

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Step 5.3.1

Factor $2 x y'$ out of $6 x^{2} y y'$ .

$2 x y' (3 x y) + 4 x y' = 8 x - 4 y - 6 y^{2} x$

Step 5.3.2

Factor $2 x y'$ out of $4 x y'$ .

$2 x y' (3 x y) + 2 x y' \cdot 2 = 8 x - 4 y - 6 y^{2} x$

Step 5.3.3

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

$2 x y' (3 x y + 2) = 8 x - 4 y - 6 y^{2} x$

Step 5.4

Divide each term in $2 x y' (3 x y + 2) = 8 x - 4 y - 6 y^{2} x$ by $2 x (3 x y + 2)$ and simplify.

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Step 5.4.1

Divide each term in $2 x y' (3 x y + 2) = 8 x - 4 y - 6 y^{2} x$ by $2 x (3 x y + 2)$ .

$\frac{2 x y' (3 x y + 2)}{2 x (3 x y + 2)} = \frac{8 x}{2 x (3 x y + 2)} + \frac{- 4 y}{2 x (3 x y + 2)} + \frac{- 6 y^{2} x}{2 x (3 x y + 2)}$

Step 5.4.2

Simplify the left side.

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Step 5.4.2.1

Cancel the common factor of $2$ .

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Step 5.4.2.1.1

Cancel the common factor.

$\frac{2 x y' (3 x y + 2)}{2 x (3 x y + 2)} = \frac{8 x}{2 x (3 x y + 2)} + \frac{- 4 y}{2 x (3 x y + 2)} + \frac{- 6 y^{2} x}{2 x (3 x y + 2)}$

Step 5.4.2.1.2

Rewrite the expression.

$\frac{x y' (3 x y + 2)}{x (3 x y + 2)} = \frac{8 x}{2 x (3 x y + 2)} + \frac{- 4 y}{2 x (3 x y + 2)} + \frac{- 6 y^{2} x}{2 x (3 x y + 2)}$

Step 5.4.2.2

Cancel the common factor of $x$ .

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Step 5.4.2.2.1

Cancel the common factor.

$\frac{x y' (3 x y + 2)}{x (3 x y + 2)} = \frac{8 x}{2 x (3 x y + 2)} + \frac{- 4 y}{2 x (3 x y + 2)} + \frac{- 6 y^{2} x}{2 x (3 x y + 2)}$

Step 5.4.2.2.2

Rewrite the expression.

$\frac{y' (3 x y + 2)}{3 x y + 2} = \frac{8 x}{2 x (3 x y + 2)} + \frac{- 4 y}{2 x (3 x y + 2)} + \frac{- 6 y^{2} x}{2 x (3 x y + 2)}$

Step 5.4.2.3

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

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Step 5.4.2.3.1

Cancel the common factor.

$\frac{y' (3 x y + 2)}{3 x y + 2} = \frac{8 x}{2 x (3 x y + 2)} + \frac{- 4 y}{2 x (3 x y + 2)} + \frac{- 6 y^{2} x}{2 x (3 x y + 2)}$

Step 5.4.2.3.2

Divide $y'$ by $1$ .

$y' = \frac{8 x}{2 x (3 x y + 2)} + \frac{- 4 y}{2 x (3 x y + 2)} + \frac{- 6 y^{2} x}{2 x (3 x y + 2)}$

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

Cancel the common factor of $8$ and $2$ .

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Step 5.4.3.1.1.1

Factor $2$ out of $8 x$ .

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

Step 5.4.3.1.1.2

Cancel the common factors.

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Step 5.4.3.1.1.2.1

Factor $2$ out of $2 x (3 x y + 2)$ .

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

Step 5.4.3.1.1.2.2

Cancel the common factor.

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

Step 5.4.3.1.1.2.3

Rewrite the expression.

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

Step 5.4.3.1.2

Cancel the common factor of $x$ .

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Step 5.4.3.1.2.1

Cancel the common factor.

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

Step 5.4.3.1.2.2

Rewrite the expression.

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

Step 5.4.3.1.3

Cancel the common factor of $- 4$ and $2$ .

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Step 5.4.3.1.3.1

Factor $2$ out of $- 4 y$ .

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

Step 5.4.3.1.3.2

Cancel the common factors.

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Step 5.4.3.1.3.2.1

Factor $2$ out of $2 x (3 x y + 2)$ .

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

Step 5.4.3.1.3.2.2

Cancel the common factor.

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

Step 5.4.3.1.3.2.3

Rewrite the expression.

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

Step 5.4.3.1.4

Move the negative in front of the fraction.

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

Step 5.4.3.1.5

Cancel the common factor of $- 6$ and $2$ .

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Step 5.4.3.1.5.1

Factor $2$ out of $- 6 y^{2} x$ .

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

Step 5.4.3.1.5.2

Cancel the common factors.

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Step 5.4.3.1.5.2.1

Factor $2$ out of $2 x (3 x y + 2)$ .

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

Step 5.4.3.1.5.2.2

Cancel the common factor.

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

Step 5.4.3.1.5.2.3

Rewrite the expression.

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

Step 5.4.3.1.6

Cancel the common factor of $x$ .

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Step 5.4.3.1.6.1

Cancel the common factor.

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

Step 5.4.3.1.6.2

Rewrite the expression.

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

Step 5.4.3.1.7

Move the negative in front of the fraction.

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

Step 5.4.3.2

To write $\frac{4}{3 x y + 2}$ as a fraction with a common denominator, multiply by $\frac{x}{x}$ .

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

Step 5.4.3.3

Write each expression with a common denominator of $(3 x y + 2) x$ , by multiplying each by an appropriate factor of $1$ .

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Step 5.4.3.3.1

Multiply $\frac{4}{3 x y + 2}$ by $\frac{x}{x}$ .

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

Step 5.4.3.3.2

Reorder the factors of $(3 x y + 2) x$ .

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

Step 5.4.3.4

Combine the numerators over the common denominator.

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

Step 5.4.3.5

Factor $2$ out of $4 x - 2 y$ .

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Step 5.4.3.5.1

Factor $2$ out of $4 x$ .

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

Step 5.4.3.5.2

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

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

Step 5.4.3.5.3

Factor $2$ out of $2 (2 x) + 2 (- y)$ .

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

Step 5.4.3.6

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

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

Step 5.4.3.7

Write each expression with a common denominator of $x (3 x y + 2)$ , by multiplying each by an appropriate factor of $1$ .

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Step 5.4.3.7.1

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

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

Step 5.4.3.7.2

Reorder the factors of $(3 x y + 2) x$ .

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

Step 5.4.3.8

Combine the numerators over the common denominator.

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

Step 5.4.3.9

Simplify the numerator.

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Step 5.4.3.9.1

Apply the distributive property.

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

Step 5.4.3.9.2

Multiply $2$ by $2$ .

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

Step 5.4.3.9.3

Multiply $- 1$ by $2$ .

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

Step 6

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

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