Calculus Examples

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

Step 1

Differentiate both sides of the equation.

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

Step 2

Differentiate the left side of the equation.

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Step 2.1

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

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

Step 2.2

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

$2 (3 x^{2})$

Step 2.3

Multiply $3$ by $2$ .

$6 x^{2}$

Step 3

Differentiate the right side of the equation.

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Step 3.1

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

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

Step 3.2

Expand $(3 x y + 1) (3 x y + 1)$ using the FOIL Method.

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Step 3.2.1

Apply the distributive property.

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

Step 3.2.2

Apply the distributive property.

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

Step 3.2.3

Apply the distributive property.

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

Step 3.3

Simplify and combine like terms.

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Step 3.3.1

Simplify each term.

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Step 3.3.1.1

Multiply $x$ by $x$ by adding the exponents.

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Step 3.3.1.1.1

Move $x$ .

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

Step 3.3.1.1.2

Multiply $x$ by $x$ .

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

Step 3.3.1.2

Multiply $y$ by $y$ by adding the exponents.

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Step 3.3.1.2.1

Move $y$ .

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

Step 3.3.1.2.2

Multiply $y$ by $y$ .

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

Step 3.3.1.3

Multiply $3$ by $3$ .

$\frac{d}{d x} [9 x^{2} y^{2} + 3 x y \cdot 1 + 1 (3 x y) + 1 \cdot 1]$

Step 3.3.1.4

Multiply $3$ by $1$ .

$\frac{d}{d x} [9 x^{2} y^{2} + 3 x y + 1 (3 x y) + 1 \cdot 1]$

Step 3.3.1.5

Multiply $3 x y$ by $1$ .

$\frac{d}{d x} [9 x^{2} y^{2} + 3 x y + 3 x y + 1 \cdot 1]$

Step 3.3.1.6

Multiply $1$ by $1$ .

$\frac{d}{d x} [9 x^{2} y^{2} + 3 x y + 3 x y + 1]$

Step 3.3.2

Add $3 x y$ and $3 x y$ .

$\frac{d}{d x} [9 x^{2} y^{2} + 6 x y + 1]$

Step 3.4

Differentiate.

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Step 3.4.1

By the Sum Rule, the derivative of $9 x^{2} y^{2} + 6 x y + 1$ with respect to $x$ is $\frac{d}{d x} [9 x^{2} y^{2}] + \frac{d}{d x} [6 x y] + \frac{d}{d x} [1]$ .

$\frac{d}{d x} [9 x^{2} y^{2}] + \frac{d}{d x} [6 x y] + \frac{d}{d x} [1]$

Step 3.4.2

Since $9$ is constant with respect to $x$ , the derivative of $9 x^{2} y^{2}$ with respect to $x$ is $9 \frac{d}{d x} [x^{2} y^{2}]$ .

$9 \frac{d}{d x} [x^{2} y^{2}] + \frac{d}{d x} [6 x y] + \frac{d}{d x} [1]$

Step 3.5

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

$9 (x^{2} \frac{d}{d x} [y^{2}] + y^{2} \frac{d}{d x} [x^{2}]) + \frac{d}{d x} [6 x y] + \frac{d}{d x} [1]$

Step 3.6

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 3.6.1

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

$9 (x^{2} (\frac{d}{d u} [u^{2}] \frac{d}{d x} [y]) + y^{2} \frac{d}{d x} [x^{2}]) + \frac{d}{d x} [6 x y] + \frac{d}{d x} [1]$

Step 3.6.2

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

$9 (x^{2} (2 u \frac{d}{d x} [y]) + y^{2} \frac{d}{d x} [x^{2}]) + \frac{d}{d x} [6 x y] + \frac{d}{d x} [1]$

Step 3.6.3

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

$9 (x^{2} (2 y \frac{d}{d x} [y]) + y^{2} \frac{d}{d x} [x^{2}]) + \frac{d}{d x} [6 x y] + \frac{d}{d x} [1]$

Step 3.7

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

$9 (2 \cdot x^{2} y \frac{d}{d x} [y] + y^{2} \frac{d}{d x} [x^{2}]) + \frac{d}{d x} [6 x y] + \frac{d}{d x} [1]$

Step 3.8

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

$9 (2 x^{2} y y' + y^{2} \frac{d}{d x} [x^{2}]) + \frac{d}{d x} [6 x y] + \frac{d}{d x} [1]$

Step 3.9

Differentiate.

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Step 3.9.1

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

$9 (2 x^{2} y y' + y^{2} (2 x)) + \frac{d}{d x} [6 x y] + \frac{d}{d x} [1]$

Step 3.9.2

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

$9 (2 x^{2} y y' + 2 \cdot y^{2} x) + \frac{d}{d x} [6 x y] + \frac{d}{d x} [1]$

Step 3.9.3

Since $6$ is constant with respect to $x$ , the derivative of $6 x y$ with respect to $x$ is $6 \frac{d}{d x} [x y]$ .

$9 (2 x^{2} y y' + 2 y^{2} x) + 6 \frac{d}{d x} [x y] + \frac{d}{d x} [1]$

Step 3.10

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 $g (x) = y$ .

$9 (2 x^{2} y y' + 2 y^{2} x) + 6 (x \frac{d}{d x} [y] + y \frac{d}{d x} [x]) + \frac{d}{d x} [1]$

Step 3.11

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

$9 (2 x^{2} y y' + 2 y^{2} x) + 6 (x y' + y \frac{d}{d x} [x]) + \frac{d}{d x} [1]$

Step 3.12

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

$9 (2 x^{2} y y' + 2 y^{2} x) + 6 (x y' + y \cdot 1) + \frac{d}{d x} [1]$

Step 3.13

Multiply $y$ by $1$ .

$9 (2 x^{2} y y' + 2 y^{2} x) + 6 (x y' + y) + \frac{d}{d x} [1]$

Step 3.14

Since $1$ is constant with respect to $x$ , the derivative of $1$ with respect to $x$ is $0$ .

$9 (2 x^{2} y y' + 2 y^{2} x) + 6 (x y' + y) + 0$

Step 3.15

Add $9 (2 x^{2} y y' + 2 y^{2} x) + 6 (x y' + y)$ and $0$ .

$9 (2 x^{2} y y' + 2 y^{2} x) + 6 (x y' + y)$

Step 3.16

Simplify.

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Step 3.16.1

Apply the distributive property.

$9 (2 x^{2} y y') + 9 (2 y^{2} x) + 6 (x y' + y)$

Step 3.16.2

Apply the distributive property.

$9 (2 x^{2} y y') + 9 (2 y^{2} x) + 6 (x y') + 6 y$

Step 3.16.3

Combine terms.

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Step 3.16.3.1

Multiply $2$ by $9$ .

$18 (x^{2} y y') + 9 (2 y^{2} x) + 6 (x y') + 6 y$

Step 3.16.3.2

Multiply $2$ by $9$ .

$18 x^{2} y y' + 18 y^{2} x + 6 x y' + 6 y$

Step 4

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

$6 x^{2} = 18 x^{2} y y' + 18 y^{2} x + 6 x y' + 6 y$

Step 5

Solve for $y'$ .

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Step 5.1

Rewrite the equation as $18 x^{2} y y' + 18 y^{2} x + 6 x y' + 6 y = 6 x^{2}$ .

$18 x^{2} y y' + 18 y^{2} x + 6 x y' + 6 y = 6 x^{2}$

Step 5.2

Move all terms not containing $y'$ to the right side of the equation.

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Step 5.2.1

Subtract $18 y^{2} x$ from both sides of the equation.

$18 x^{2} y y' + 6 x y' + 6 y = 6 x^{2} - 18 y^{2} x$

Step 5.2.2

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

$18 x^{2} y y' + 6 x y' = 6 x^{2} - 18 y^{2} x - 6 y$

Step 5.3

Factor $6 x y'$ out of $18 x^{2} y y' + 6 x y'$ .

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Step 5.3.1

Factor $6 x y'$ out of $18 x^{2} y y'$ .

$6 x y' (3 x y) + 6 x y' = 6 x^{2} - 18 y^{2} x - 6 y$

Step 5.3.2

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

$6 x y' (3 x y) + 6 x y' \cdot 1 = 6 x^{2} - 18 y^{2} x - 6 y$

Step 5.3.3

Factor $6 x y'$ out of $6 x y' (3 x y) + 6 x y' \cdot 1$ .

$6 x y' (3 x y + 1) = 6 x^{2} - 18 y^{2} x - 6 y$

Step 5.4

Divide each term in $6 x y' (3 x y + 1) = 6 x^{2} - 18 y^{2} x - 6 y$ by $6 x (3 x y + 1)$ and simplify.

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Step 5.4.1

Divide each term in $6 x y' (3 x y + 1) = 6 x^{2} - 18 y^{2} x - 6 y$ by $6 x (3 x y + 1)$ .

$\frac{6 x y' (3 x y + 1)}{6 x (3 x y + 1)} = \frac{6 x^{2}}{6 x (3 x y + 1)} + \frac{- 18 y^{2} x}{6 x (3 x y + 1)} + \frac{- 6 y}{6 x (3 x y + 1)}$

Step 5.4.2

Simplify the left side.

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Step 5.4.2.1

Cancel the common factor of $6$ .

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Step 5.4.2.1.1

Cancel the common factor.

$\frac{6 x y' (3 x y + 1)}{6 x (3 x y + 1)} = \frac{6 x^{2}}{6 x (3 x y + 1)} + \frac{- 18 y^{2} x}{6 x (3 x y + 1)} + \frac{- 6 y}{6 x (3 x y + 1)}$

Step 5.4.2.1.2

Rewrite the expression.

$\frac{x y' (3 x y + 1)}{x (3 x y + 1)} = \frac{6 x^{2}}{6 x (3 x y + 1)} + \frac{- 18 y^{2} x}{6 x (3 x y + 1)} + \frac{- 6 y}{6 x (3 x y + 1)}$

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.

$\frac{x y' (3 x y + 1)}{x (3 x y + 1)} = \frac{6 x^{2}}{6 x (3 x y + 1)} + \frac{- 18 y^{2} x}{6 x (3 x y + 1)} + \frac{- 6 y}{6 x (3 x y + 1)}$

Step 5.4.2.2.2

Rewrite the expression.

$\frac{y' (3 x y + 1)}{3 x y + 1} = \frac{6 x^{2}}{6 x (3 x y + 1)} + \frac{- 18 y^{2} x}{6 x (3 x y + 1)} + \frac{- 6 y}{6 x (3 x y + 1)}$

Step 5.4.2.3

Cancel the common factor of $3 x y + 1$ .

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Step 5.4.2.3.1

Cancel the common factor.

$\frac{y' (3 x y + 1)}{3 x y + 1} = \frac{6 x^{2}}{6 x (3 x y + 1)} + \frac{- 18 y^{2} x}{6 x (3 x y + 1)} + \frac{- 6 y}{6 x (3 x y + 1)}$

Step 5.4.2.3.2

Divide $y'$ by $1$ .

$y' = \frac{6 x^{2}}{6 x (3 x y + 1)} + \frac{- 18 y^{2} x}{6 x (3 x y + 1)} + \frac{- 6 y}{6 x (3 x y + 1)}$

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 $6$ .

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Step 5.4.3.1.1.1

Cancel the common factor.

$y' = \frac{6 x^{2}}{6 x (3 x y + 1)} + \frac{- 18 y^{2} x}{6 x (3 x y + 1)} + \frac{- 6 y}{6 x (3 x y + 1)}$

Step 5.4.3.1.1.2

Rewrite the expression.

$y' = \frac{x^{2}}{x (3 x y + 1)} + \frac{- 18 y^{2} x}{6 x (3 x y + 1)} + \frac{- 6 y}{6 x (3 x y + 1)}$

Step 5.4.3.1.2

Cancel the common factor of $x^{2}$ and $x$ .

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Step 5.4.3.1.2.1

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

$y' = \frac{x \cdot x}{x (3 x y + 1)} + \frac{- 18 y^{2} x}{6 x (3 x y + 1)} + \frac{- 6 y}{6 x (3 x y + 1)}$

Step 5.4.3.1.2.2

Cancel the common factors.

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Step 5.4.3.1.2.2.1

Cancel the common factor.

$y' = \frac{x \cdot x}{x (3 x y + 1)} + \frac{- 18 y^{2} x}{6 x (3 x y + 1)} + \frac{- 6 y}{6 x (3 x y + 1)}$

Step 5.4.3.1.2.2.2

Rewrite the expression.

$y' = \frac{x}{3 x y + 1} + \frac{- 18 y^{2} x}{6 x (3 x y + 1)} + \frac{- 6 y}{6 x (3 x y + 1)}$

Step 5.4.3.1.3

Cancel the common factor of $- 18$ and $6$ .

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Step 5.4.3.1.3.1

Factor $6$ out of $- 18 y^{2} x$ .

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

Step 5.4.3.1.3.2

Cancel the common factors.

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Step 5.4.3.1.3.2.1

Factor $6$ out of $6 x (3 x y + 1)$ .

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

Step 5.4.3.1.3.2.2

Cancel the common factor.

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

Step 5.4.3.1.3.2.3

Rewrite the expression.

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

Step 5.4.3.1.4

Cancel the common factor of $x$ .

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Step 5.4.3.1.4.1

Cancel the common factor.

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

Step 5.4.3.1.4.2

Rewrite the expression.

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

Step 5.4.3.1.5

Move the negative in front of the fraction.

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

Step 5.4.3.1.6

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

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Step 5.4.3.1.6.1

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

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

Step 5.4.3.1.6.2

Cancel the common factors.

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Step 5.4.3.1.6.2.1

Factor $6$ out of $6 x (3 x y + 1)$ .

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

Step 5.4.3.1.6.2.2

Cancel the common factor.

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

Step 5.4.3.1.6.2.3

Rewrite the expression.

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

Step 5.4.3.1.7

Move the negative in front of the fraction.

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

Step 5.4.3.2

Combine the numerators over the common denominator.

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

Step 5.4.3.3

To write $\frac{x - 3 y^{2}}{3 x y + 1}$ as a fraction with a common denominator, multiply by $\frac{x}{x}$ .

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

Step 5.4.3.4

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

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Step 5.4.3.4.1

Multiply $\frac{x - 3 y^{2}}{3 x y + 1}$ by $\frac{x}{x}$ .

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

Step 5.4.3.4.2

Reorder the factors of $(3 x y + 1) x$ .

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

Step 5.4.3.5

Combine the numerators over the common denominator.

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

Step 5.4.3.6

Simplify the numerator.

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Step 5.4.3.6.1

Apply the distributive property.

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

Step 5.4.3.6.2

Multiply $x$ by $x$ .

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

Step 6

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

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