Calculus Examples

${(3 x y + 7)}^{2} = 6 y$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} ({(3 x y + 7)}^{2}) = \frac{d}{d x} (6 y)$

Step 2

Differentiate the left side of the equation.

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Step 2.1

Rewrite ${(3 x y + 7)}^{2}$ as $(3 x y + 7) (3 x y + 7)$ .

$\frac{d}{d x} [(3 x y + 7) (3 x y + 7)]$

Step 2.2

Expand $(3 x y + 7) (3 x y + 7)$ using the FOIL Method.

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Step 2.2.1

Apply the distributive property.

$\frac{d}{d x} [3 x y (3 x y + 7) + 7 (3 x y + 7)]$

Step 2.2.2

Apply the distributive property.

$\frac{d}{d x} [3 x y (3 x y) + 3 x y \cdot 7 + 7 (3 x y + 7)]$

Step 2.2.3

Apply the distributive property.

$\frac{d}{d x} [3 x y (3 x y) + 3 x y \cdot 7 + 7 (3 x y) + 7 \cdot 7]$

Step 2.3

Simplify and combine like terms.

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Step 2.3.1

Simplify each term.

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Step 2.3.1.1

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

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Step 2.3.1.1.1

Move $x$ .

$\frac{d}{d x} [3 (x \cdot x) y (3 y) + 3 x y \cdot 7 + 7 (3 x y) + 7 \cdot 7]$

Step 2.3.1.1.2

Multiply $x$ by $x$ .

$\frac{d}{d x} [3 x^{2} y (3 y) + 3 x y \cdot 7 + 7 (3 x y) + 7 \cdot 7]$

Step 2.3.1.2

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

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Step 2.3.1.2.1

Move $y$ .

$\frac{d}{d x} [3 x^{2} (y \cdot y) \cdot 3 + 3 x y \cdot 7 + 7 (3 x y) + 7 \cdot 7]$

Step 2.3.1.2.2

Multiply $y$ by $y$ .

$\frac{d}{d x} [3 x^{2} y^{2} \cdot 3 + 3 x y \cdot 7 + 7 (3 x y) + 7 \cdot 7]$

Step 2.3.1.3

Multiply $3$ by $3$ .

$\frac{d}{d x} [9 x^{2} y^{2} + 3 x y \cdot 7 + 7 (3 x y) + 7 \cdot 7]$

Step 2.3.1.4

Multiply $7$ by $3$ .

$\frac{d}{d x} [9 x^{2} y^{2} + 21 x y + 7 (3 x y) + 7 \cdot 7]$

Step 2.3.1.5

Multiply $3$ by $7$ .

$\frac{d}{d x} [9 x^{2} y^{2} + 21 x y + 21 (x y) + 7 \cdot 7]$

Step 2.3.1.6

Multiply $7$ by $7$ .

$\frac{d}{d x} [9 x^{2} y^{2} + 21 x y + 21 x y + 49]$

Step 2.3.2

Add $21 x y$ and $21 x y$ .

$\frac{d}{d x} [9 x^{2} y^{2} + 42 x y + 49]$

Step 2.4

Differentiate.

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Step 2.4.1

By the Sum Rule, the derivative of $9 x^{2} y^{2} + 42 x y + 49$ with respect to $x$ is $\frac{d}{d x} [9 x^{2} y^{2}] + \frac{d}{d x} [42 x y] + \frac{d}{d x} [49]$ .

$\frac{d}{d x} [9 x^{2} y^{2}] + \frac{d}{d x} [42 x y] + \frac{d}{d x} [49]$

Step 2.4.2

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

$9 \frac{d}{d x} [x^{2} y^{2}] + \frac{d}{d x} [42 x y] + \frac{d}{d x} [49]$

Step 2.5

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

$9 (x^{2} \frac{d}{d x} [y^{2}] + y^{2} \frac{d}{d x} [x^{2}]) + \frac{d}{d x} [42 x y] + \frac{d}{d x} [49]$

Step 2.6

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 2.6.1

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

$9 (x^{2} (\frac{d}{d u} [u^{2}] \frac{d}{d x} [y]) + y^{2} \frac{d}{d x} [x^{2}]) + \frac{d}{d x} [42 x y] + \frac{d}{d x} [49]$

Step 2.6.2

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

$9 (x^{2} (2 u \frac{d}{d x} [y]) + y^{2} \frac{d}{d x} [x^{2}]) + \frac{d}{d x} [42 x y] + \frac{d}{d x} [49]$

Step 2.6.3

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

$9 (x^{2} (2 y \frac{d}{d x} [y]) + y^{2} \frac{d}{d x} [x^{2}]) + \frac{d}{d x} [42 x y] + \frac{d}{d x} [49]$

Step 2.7

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

$9 (2 \cdot x^{2} y \frac{d}{d x} [y] + y^{2} \frac{d}{d x} [x^{2}]) + \frac{d}{d x} [42 x y] + \frac{d}{d x} [49]$

Step 2.8

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

$9 (2 x^{2} y y' + y^{2} \frac{d}{d x} [x^{2}]) + \frac{d}{d x} [42 x y] + \frac{d}{d x} [49]$

Step 2.9

Differentiate.

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Step 2.9.1

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

$9 (2 x^{2} y y' + y^{2} (2 x)) + \frac{d}{d x} [42 x y] + \frac{d}{d x} [49]$

Step 2.9.2

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

$9 (2 x^{2} y y' + 2 \cdot y^{2} x) + \frac{d}{d x} [42 x y] + \frac{d}{d x} [49]$

Step 2.9.3

Since $42$ is constant with respect to $x$ , the derivative of $42 x y$ with respect to $x$ is $42 \frac{d}{d x} [x y]$ .

$9 (2 x^{2} y y' + 2 y^{2} x) + 42 \frac{d}{d x} [x y] + \frac{d}{d x} [49]$

Step 2.10

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 $g (x) = y$ .

$9 (2 x^{2} y y' + 2 y^{2} x) + 42 (x \frac{d}{d x} [y] + y \frac{d}{d x} [x]) + \frac{d}{d x} [49]$

Step 2.11

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

$9 (2 x^{2} y y' + 2 y^{2} x) + 42 (x y' + y \frac{d}{d x} [x]) + \frac{d}{d x} [49]$

Step 2.12

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

$9 (2 x^{2} y y' + 2 y^{2} x) + 42 (x y' + y \cdot 1) + \frac{d}{d x} [49]$

Step 2.13

Multiply $y$ by $1$ .

$9 (2 x^{2} y y' + 2 y^{2} x) + 42 (x y' + y) + \frac{d}{d x} [49]$

Step 2.14

Since $49$ is constant with respect to $x$ , the derivative of $49$ with respect to $x$ is $0$ .

$9 (2 x^{2} y y' + 2 y^{2} x) + 42 (x y' + y) + 0$

Step 2.15

Add $9 (2 x^{2} y y' + 2 y^{2} x) + 42 (x y' + y)$ and $0$ .

$9 (2 x^{2} y y' + 2 y^{2} x) + 42 (x y' + y)$

Step 2.16

Simplify.

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Step 2.16.1

Apply the distributive property.

$9 (2 x^{2} y y') + 9 (2 y^{2} x) + 42 (x y' + y)$

Step 2.16.2

Apply the distributive property.

$9 (2 x^{2} y y') + 9 (2 y^{2} x) + 42 (x y') + 42 y$

Step 2.16.3

Combine terms.

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Step 2.16.3.1

Multiply $2$ by $9$ .

$18 (x^{2} y y') + 9 (2 y^{2} x) + 42 (x y') + 42 y$

Step 2.16.3.2

Multiply $2$ by $9$ .

$18 x^{2} y y' + 18 y^{2} x + 42 x y' + 42 y$

Step 3

Differentiate the right side of the equation.

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Step 3.1

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

$6 \frac{d}{d x} [y]$

Step 3.2

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

$6 y'$

Step 4

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

$18 x^{2} y y' + 18 y^{2} x + 42 x y' + 42 y = 6 y'$

Step 5

Solve for $y'$ .

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Step 5.1

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

$18 x^{2} y y' + 18 y^{2} x + 42 x y' + 42 y - 6 y' = 0$

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.

$18 x^{2} y y' + 42 x y' + 42 y - 6 y' = - 18 y^{2} x$

Step 5.2.2

Subtract $42 y$ from both sides of the equation.

$18 x^{2} y y' + 42 x y' - 6 y' = - 18 y^{2} x - 42 y$

Step 5.3

Factor $6 y'$ out of $18 x^{2} y y' + 42 x y' - 6 y'$ .

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Step 5.3.1

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

$6 y' (3 x^{2} y) + 42 x y' - 6 y' = - 18 y^{2} x - 42 y$

Step 5.3.2

Factor $6 y'$ out of $42 x y'$ .

$6 y' (3 x^{2} y) + 6 y' (7 x) - 6 y' = - 18 y^{2} x - 42 y$

Step 5.3.3

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

$6 y' (3 x^{2} y) + 6 y' (7 x) + 6 y' \cdot - 1 = - 18 y^{2} x - 42 y$

Step 5.3.4

Factor $6 y'$ out of $6 y' (3 x^{2} y) + 6 y' (7 x)$ .

$6 y' (3 x^{2} y + 7 x) + 6 y' \cdot - 1 = - 18 y^{2} x - 42 y$

Step 5.3.5

Factor $6 y'$ out of $6 y' (3 x^{2} y + 7 x) + 6 y' \cdot - 1$ .

$6 y' (3 x^{2} y + 7 x - 1) = - 18 y^{2} x - 42 y$

Step 5.4

Divide each term in $6 y' (3 x^{2} y + 7 x - 1) = - 18 y^{2} x - 42 y$ by $6 (3 x^{2} y + 7 x - 1)$ and simplify.

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Step 5.4.1

Divide each term in $6 y' (3 x^{2} y + 7 x - 1) = - 18 y^{2} x - 42 y$ by $6 (3 x^{2} y + 7 x - 1)$ .

$\frac{6 y' (3 x^{2} y + 7 x - 1)}{6 (3 x^{2} y + 7 x - 1)} = \frac{- 18 y^{2} x}{6 (3 x^{2} y + 7 x - 1)} + \frac{- 42 y}{6 (3 x^{2} y + 7 x - 1)}$

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.

$\frac{6 y' (3 x^{2} y + 7 x - 1)}{6 (3 x^{2} y + 7 x - 1)} = \frac{- 18 y^{2} x}{6 (3 x^{2} y + 7 x - 1)} + \frac{- 42 y}{6 (3 x^{2} y + 7 x - 1)}$

Step 5.4.2.1.2

Rewrite the expression.

$\frac{y' (3 x^{2} y + 7 x - 1)}{3 x^{2} y + 7 x - 1} = \frac{- 18 y^{2} x}{6 (3 x^{2} y + 7 x - 1)} + \frac{- 42 y}{6 (3 x^{2} y + 7 x - 1)}$

Step 5.4.2.2

Cancel the common factor of $3 x^{2} y + 7 x - 1$ .

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Step 5.4.2.2.1

Cancel the common factor.

$\frac{y' (3 x^{2} y + 7 x - 1)}{3 x^{2} y + 7 x - 1} = \frac{- 18 y^{2} x}{6 (3 x^{2} y + 7 x - 1)} + \frac{- 42 y}{6 (3 x^{2} y + 7 x - 1)}$

Step 5.4.2.2.2

Divide $y'$ by $1$ .

$y' = \frac{- 18 y^{2} x}{6 (3 x^{2} y + 7 x - 1)} + \frac{- 42 y}{6 (3 x^{2} y + 7 x - 1)}$

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 $- 18$ and $6$ .

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Step 5.4.3.1.1.1

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

$y' = \frac{6 (- 3 y^{2} x)}{6 (3 x^{2} y + 7 x - 1)} + \frac{- 42 y}{6 (3 x^{2} y + 7 x - 1)}$

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.

$y' = \frac{6 (- 3 y^{2} x)}{6 (3 x^{2} y + 7 x - 1)} + \frac{- 42 y}{6 (3 x^{2} y + 7 x - 1)}$

Step 5.4.3.1.1.2.2

Rewrite the expression.

$y' = \frac{- 3 y^{2} x}{3 x^{2} y + 7 x - 1} + \frac{- 42 y}{6 (3 x^{2} y + 7 x - 1)}$

Step 5.4.3.1.2

Move the negative in front of the fraction.

$y' = - \frac{3 y^{2} x}{3 x^{2} y + 7 x - 1} + \frac{- 42 y}{6 (3 x^{2} y + 7 x - 1)}$

Step 5.4.3.1.3

Cancel the common factor of $- 42$ and $6$ .

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Step 5.4.3.1.3.1

Factor $6$ out of $- 42 y$ .

$y' = - \frac{3 y^{2} x}{3 x^{2} y + 7 x - 1} + \frac{6 (- 7 y)}{6 (3 x^{2} y + 7 x - 1)}$

Step 5.4.3.1.3.2

Cancel the common factors.

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Step 5.4.3.1.3.2.1

Cancel the common factor.

$y' = - \frac{3 y^{2} x}{3 x^{2} y + 7 x - 1} + \frac{6 (- 7 y)}{6 (3 x^{2} y + 7 x - 1)}$

Step 5.4.3.1.3.2.2

Rewrite the expression.

$y' = - \frac{3 y^{2} x}{3 x^{2} y + 7 x - 1} + \frac{- 7 y}{3 x^{2} y + 7 x - 1}$

Step 5.4.3.1.4

Move the negative in front of the fraction.

$y' = - \frac{3 y^{2} x}{3 x^{2} y + 7 x - 1} - \frac{7 y}{3 x^{2} y + 7 x - 1}$

Step 5.4.3.2

Simplify terms.

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Step 5.4.3.2.1

Combine the numerators over the common denominator.

$y' = \frac{- 3 y^{2} x - 7 y}{3 x^{2} y + 7 x - 1}$

Step 5.4.3.2.2

Factor $y$ out of $- 3 y^{2} x - 7 y$ .

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Step 5.4.3.2.2.1

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

$y' = \frac{y (- 3 y x) - 7 y}{3 x^{2} y + 7 x - 1}$

Step 5.4.3.2.2.2

Factor $y$ out of $- 7 y$ .

$y' = \frac{y (- 3 y x) + y \cdot - 7}{3 x^{2} y + 7 x - 1}$

Step 5.4.3.2.2.3

Factor $y$ out of $y (- 3 y x) + y \cdot - 7$ .

$y' = \frac{y (- 3 y x - 7)}{3 x^{2} y + 7 x - 1}$

Step 5.4.3.2.3

Factor $- 1$ out of $- 3 y x$ .

$y' = \frac{y (- (3 y x) - 7)}{3 x^{2} y + 7 x - 1}$

Step 5.4.3.2.4

Rewrite $- 7$ as $- 1 (7)$ .

$y' = \frac{y (- (3 y x) - 1 (7))}{3 x^{2} y + 7 x - 1}$

Step 5.4.3.2.5

Factor $- 1$ out of $- (3 y x) - 1 (7)$ .

$y' = \frac{y (- (3 y x + 7))}{3 x^{2} y + 7 x - 1}$

Step 5.4.3.2.6

Simplify the expression.

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Step 5.4.3.2.6.1

Rewrite $- (3 y x + 7)$ as $- 1 (3 y x + 7)$ .

$y' = \frac{y (- 1 (3 y x + 7))}{3 x^{2} y + 7 x - 1}$

Step 5.4.3.2.6.2

Move the negative in front of the fraction.

$y' = - \frac{y (3 y x + 7)}{3 x^{2} y + 7 x - 1}$

Step 6

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

$\frac{d y}{d x} = - \frac{y (3 y x + 7)}{3 x^{2} y + 7 x - 1}$