Calculus Examples

${(6 x y + 5)}^{2} = 12 y$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d y} ({(6 x y + 5)}^{2}) = \frac{d}{d y} (12 y)$

Step 2

Differentiate the left side of the equation.

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Step 2.1

Rewrite ${(6 x y + 5)}^{2}$ as $(6 x y + 5) (6 x y + 5)$ .

$\frac{d}{d y} [(6 x y + 5) (6 x y + 5)]$

Step 2.2

Expand $(6 x y + 5) (6 x y + 5)$ using the FOIL Method.

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Step 2.2.1

Apply the distributive property.

$\frac{d}{d y} [6 x y (6 x y + 5) + 5 (6 x y + 5)]$

Step 2.2.2

Apply the distributive property.

$\frac{d}{d y} [6 x y (6 x y) + 6 x y \cdot 5 + 5 (6 x y + 5)]$

Step 2.2.3

Apply the distributive property.

$\frac{d}{d y} [6 x y (6 x y) + 6 x y \cdot 5 + 5 (6 x y) + 5 \cdot 5]$

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 $x$ .

$\frac{d}{d y} [6 (x \cdot x) y (6 y) + 6 x y \cdot 5 + 5 (6 x y) + 5 \cdot 5]$

Step 2.3.1.1.2

Multiply $x$ by $x$ .

$\frac{d}{d y} [6 x^{2} y (6 y) + 6 x y \cdot 5 + 5 (6 x y) + 5 \cdot 5]$

Step 2.3.1.2

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

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Step 2.3.1.2.1

Move $y$ .

$\frac{d}{d y} [6 x^{2} (y \cdot y) \cdot 6 + 6 x y \cdot 5 + 5 (6 x y) + 5 \cdot 5]$

Step 2.3.1.2.2

Multiply $y$ by $y$ .

$\frac{d}{d y} [6 x^{2} y^{2} \cdot 6 + 6 x y \cdot 5 + 5 (6 x y) + 5 \cdot 5]$

Step 2.3.1.3

Multiply $6$ by $6$ .

$\frac{d}{d y} [36 x^{2} y^{2} + 6 x y \cdot 5 + 5 (6 x y) + 5 \cdot 5]$

Step 2.3.1.4

Multiply $5$ by $6$ .

$\frac{d}{d y} [36 x^{2} y^{2} + 30 x y + 5 (6 x y) + 5 \cdot 5]$

Step 2.3.1.5

Multiply $6$ by $5$ .

$\frac{d}{d y} [36 x^{2} y^{2} + 30 x y + 30 (x y) + 5 \cdot 5]$

Step 2.3.1.6

Multiply $5$ by $5$ .

$\frac{d}{d y} [36 x^{2} y^{2} + 30 x y + 30 x y + 25]$

Step 2.3.2

Add $30 x y$ and $30 x y$ .

$\frac{d}{d y} [36 x^{2} y^{2} + 60 x y + 25]$

Step 2.4

Differentiate.

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Step 2.4.1

By the Sum Rule, the derivative of $36 x^{2} y^{2} + 60 x y + 25$ with respect to $y$ is $\frac{d}{d y} [36 x^{2} y^{2}] + \frac{d}{d y} [60 x y] + \frac{d}{d y} [25]$ .

$\frac{d}{d y} [36 x^{2} y^{2}] + \frac{d}{d y} [60 x y] + \frac{d}{d y} [25]$

Step 2.4.2

Since $36$ is constant with respect to $y$ , the derivative of $36 x^{2} y^{2}$ with respect to $y$ is $36 \frac{d}{d y} [x^{2} y^{2}]$ .

$36 \frac{d}{d y} [x^{2} y^{2}] + \frac{d}{d y} [60 x y] + \frac{d}{d y} [25]$

Step 2.5

Differentiate using the Product Rule which states that $\frac{d}{d y} [f (y) g (y)]$ is $f (y) \frac{d}{d y} [g (y)] + g (y) \frac{d}{d y} [f (y)]$ where $f (y) = x^{2}$ and $g (y) = y^{2}$ .

$36 (x^{2} \frac{d}{d y} [y^{2}] + y^{2} \frac{d}{d y} [x^{2}]) + \frac{d}{d y} [60 x y] + \frac{d}{d y} [25]$

Step 2.6

Differentiate using the Power Rule.

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Step 2.6.1

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

$36 (x^{2} (2 y) + y^{2} \frac{d}{d y} [x^{2}]) + \frac{d}{d y} [60 x y] + \frac{d}{d y} [25]$

Step 2.6.2

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

$36 (2 \cdot x^{2} y + y^{2} \frac{d}{d y} [x^{2}]) + \frac{d}{d y} [60 x y] + \frac{d}{d y} [25]$

Step 2.7

Differentiate using the chain rule, which states that $\frac{d}{d y} [f (g (y))]$ is $f' (g (y)) g' (y)$ where $f (y) = y^{2}$ and $g (y) = x$ .

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Step 2.7.1

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

$36 (2 x^{2} y + y^{2} (\frac{d}{d u} [u^{2}] \frac{d}{d y} [x])) + \frac{d}{d y} [60 x y] + \frac{d}{d y} [25]$

Step 2.7.2

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

$36 (2 x^{2} y + y^{2} (2 u \frac{d}{d y} [x])) + \frac{d}{d y} [60 x y] + \frac{d}{d y} [25]$

Step 2.7.3

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

$36 (2 x^{2} y + y^{2} (2 x \frac{d}{d y} [x])) + \frac{d}{d y} [60 x y] + \frac{d}{d y} [25]$

Step 2.8

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

$36 (2 x^{2} y + 2 \cdot y^{2} x \frac{d}{d y} [x]) + \frac{d}{d y} [60 x y] + \frac{d}{d y} [25]$

Step 2.9

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

$36 (2 x^{2} y + 2 y^{2} x x') + \frac{d}{d y} [60 x y] + \frac{d}{d y} [25]$

Step 2.10

Since $60$ is constant with respect to $y$ , the derivative of $60 x y$ with respect to $y$ is $60 \frac{d}{d y} [x y]$ .

$36 (2 x^{2} y + 2 y^{2} x x') + 60 \frac{d}{d y} [x y] + \frac{d}{d y} [25]$

Step 2.11

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

$36 (2 x^{2} y + 2 y^{2} x x') + 60 (x \frac{d}{d y} [y] + y \frac{d}{d y} [x]) + \frac{d}{d y} [25]$

Step 2.12

Differentiate using the Power Rule.

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Step 2.12.1

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

$36 (2 x^{2} y + 2 y^{2} x x') + 60 (x \cdot 1 + y \frac{d}{d y} [x]) + \frac{d}{d y} [25]$

Step 2.12.2

Multiply $x$ by $1$ .

$36 (2 x^{2} y + 2 y^{2} x x') + 60 (x + y \frac{d}{d y} [x]) + \frac{d}{d y} [25]$

Step 2.13

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

$36 (2 x^{2} y + 2 y^{2} x x') + 60 (x + y x') + \frac{d}{d y} [25]$

Step 2.14

Since $25$ is constant with respect to $y$ , the derivative of $25$ with respect to $y$ is $0$ .

$36 (2 x^{2} y + 2 y^{2} x x') + 60 (x + y x') + 0$

Step 2.15

Add $36 (2 x^{2} y + 2 y^{2} x x') + 60 (x + y x')$ and $0$ .

$36 (2 x^{2} y + 2 y^{2} x x') + 60 (x + y x')$

Step 2.16

Simplify.

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Step 2.16.1

Apply the distributive property.

$36 (2 x^{2} y) + 36 (2 y^{2} x x') + 60 (x + y x')$

Step 2.16.2

Apply the distributive property.

$36 (2 x^{2} y) + 36 (2 y^{2} x x') + 60 x + 60 (y x')$

Step 2.16.3

Combine terms.

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Step 2.16.3.1

Multiply $2$ by $36$ .

$72 (x^{2} y) + 36 (2 y^{2} x x') + 60 x + 60 (y x')$

Step 2.16.3.2

Multiply $2$ by $36$ .

$72 x^{2} y + 72 y^{2} x x' + 60 x + 60 y x'$

Step 2.16.4

Reorder terms.

$72 x^{2} y + 72 y^{2} x x' + 60 y x' + 60 x$

Step 3

Differentiate the right side of the equation.

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Step 3.1

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

$12 \frac{d}{d y} [y]$

Step 3.2

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

$12 \cdot 1$

Step 3.3

Multiply $12$ by $1$ .

$12$

Step 4

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

$72 x^{2} y + 72 y^{2} x x' + 60 y x' + 60 x = 12$

Step 5

Solve for $x'$ .

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Step 5.1

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

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Step 5.1.1

Subtract $72 x^{2} y$ from both sides of the equation.

$72 y^{2} x x' + 60 y x' + 60 x = 12 - 72 x^{2} y$

Step 5.1.2

Subtract $60 x$ from both sides of the equation.

$72 y^{2} x x' + 60 y x' = 12 - 72 x^{2} y - 60 x$

Step 5.2

Factor $12 y x'$ out of $72 y^{2} x x' + 60 y x'$ .

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Step 5.2.1

Factor $12 y x'$ out of $72 y^{2} x x'$ .

$12 y x' (6 y x) + 60 y x' = 12 - 72 x^{2} y - 60 x$

Step 5.2.2

Factor $12 y x'$ out of $60 y x'$ .

$12 y x' (6 y x) + 12 y x' \cdot 5 = 12 - 72 x^{2} y - 60 x$

Step 5.2.3

Factor $12 y x'$ out of $12 y x' (6 y x) + 12 y x' \cdot 5$ .

$12 y x' (6 y x + 5) = 12 - 72 x^{2} y - 60 x$

Step 5.3

Divide each term in $12 y x' (6 y x + 5) = 12 - 72 x^{2} y - 60 x$ by $12 y (6 y x + 5)$ and simplify.

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Step 5.3.1

Divide each term in $12 y x' (6 y x + 5) = 12 - 72 x^{2} y - 60 x$ by $12 y (6 y x + 5)$ .

$\frac{12 y x' (6 y x + 5)}{12 y (6 y x + 5)} = \frac{12}{12 y (6 y x + 5)} + \frac{- 72 x^{2} y}{12 y (6 y x + 5)} + \frac{- 60 x}{12 y (6 y x + 5)}$

Step 5.3.2

Simplify the left side.

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Step 5.3.2.1

Cancel the common factor of $12$ .

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Step 5.3.2.1.1

Cancel the common factor.

$\frac{12 y x' (6 y x + 5)}{12 y (6 y x + 5)} = \frac{12}{12 y (6 y x + 5)} + \frac{- 72 x^{2} y}{12 y (6 y x + 5)} + \frac{- 60 x}{12 y (6 y x + 5)}$

Step 5.3.2.1.2

Rewrite the expression.

$\frac{y x' (6 y x + 5)}{y (6 y x + 5)} = \frac{12}{12 y (6 y x + 5)} + \frac{- 72 x^{2} y}{12 y (6 y x + 5)} + \frac{- 60 x}{12 y (6 y x + 5)}$

Step 5.3.2.2

Cancel the common factor of $y$ .

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Step 5.3.2.2.1

Cancel the common factor.

$\frac{y x' (6 y x + 5)}{y (6 y x + 5)} = \frac{12}{12 y (6 y x + 5)} + \frac{- 72 x^{2} y}{12 y (6 y x + 5)} + \frac{- 60 x}{12 y (6 y x + 5)}$

Step 5.3.2.2.2

Rewrite the expression.

$\frac{x' (6 y x + 5)}{6 y x + 5} = \frac{12}{12 y (6 y x + 5)} + \frac{- 72 x^{2} y}{12 y (6 y x + 5)} + \frac{- 60 x}{12 y (6 y x + 5)}$

Step 5.3.2.3

Cancel the common factor of $6 y x + 5$ .

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Step 5.3.2.3.1

Cancel the common factor.

$\frac{x' (6 y x + 5)}{6 y x + 5} = \frac{12}{12 y (6 y x + 5)} + \frac{- 72 x^{2} y}{12 y (6 y x + 5)} + \frac{- 60 x}{12 y (6 y x + 5)}$

Step 5.3.2.3.2

Divide $x'$ by $1$ .

$x' = \frac{12}{12 y (6 y x + 5)} + \frac{- 72 x^{2} y}{12 y (6 y x + 5)} + \frac{- 60 x}{12 y (6 y x + 5)}$

Step 5.3.3

Simplify the right side.

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Step 5.3.3.1

Simplify each term.

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Step 5.3.3.1.1

Cancel the common factor of $12$ .

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Step 5.3.3.1.1.1

Cancel the common factor.

$x' = \frac{12}{12 y (6 y x + 5)} + \frac{- 72 x^{2} y}{12 y (6 y x + 5)} + \frac{- 60 x}{12 y (6 y x + 5)}$

Step 5.3.3.1.1.2

Rewrite the expression.

$x' = \frac{1}{y (6 y x + 5)} + \frac{- 72 x^{2} y}{12 y (6 y x + 5)} + \frac{- 60 x}{12 y (6 y x + 5)}$

Step 5.3.3.1.2

Cancel the common factor of $- 72$ and $12$ .

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Step 5.3.3.1.2.1

Factor $12$ out of $- 72 x^{2} y$ .

$x' = \frac{1}{y (6 y x + 5)} + \frac{12 (- 6 x^{2} y)}{12 y (6 y x + 5)} + \frac{- 60 x}{12 y (6 y x + 5)}$

Step 5.3.3.1.2.2

Cancel the common factors.

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Step 5.3.3.1.2.2.1

Factor $12$ out of $12 y (6 y x + 5)$ .

$x' = \frac{1}{y (6 y x + 5)} + \frac{12 (- 6 x^{2} y)}{12 (y (6 y x + 5))} + \frac{- 60 x}{12 y (6 y x + 5)}$

Step 5.3.3.1.2.2.2

Cancel the common factor.

$x' = \frac{1}{y (6 y x + 5)} + \frac{12 (- 6 x^{2} y)}{12 (y (6 y x + 5))} + \frac{- 60 x}{12 y (6 y x + 5)}$

Step 5.3.3.1.2.2.3

Rewrite the expression.

$x' = \frac{1}{y (6 y x + 5)} + \frac{- 6 x^{2} y}{y (6 y x + 5)} + \frac{- 60 x}{12 y (6 y x + 5)}$

Step 5.3.3.1.3

Cancel the common factor of $y$ .

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Step 5.3.3.1.3.1

Cancel the common factor.

$x' = \frac{1}{y (6 y x + 5)} + \frac{- 6 x^{2} y}{y (6 y x + 5)} + \frac{- 60 x}{12 y (6 y x + 5)}$

Step 5.3.3.1.3.2

Rewrite the expression.

$x' = \frac{1}{y (6 y x + 5)} + \frac{- 6 x^{2}}{6 y x + 5} + \frac{- 60 x}{12 y (6 y x + 5)}$

Step 5.3.3.1.4

Move the negative in front of the fraction.

$x' = \frac{1}{y (6 y x + 5)} - \frac{6 x^{2}}{6 y x + 5} + \frac{- 60 x}{12 y (6 y x + 5)}$

Step 5.3.3.1.5

Cancel the common factor of $- 60$ and $12$ .

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Step 5.3.3.1.5.1

Factor $12$ out of $- 60 x$ .

$x' = \frac{1}{y (6 y x + 5)} - \frac{6 x^{2}}{6 y x + 5} + \frac{12 (- 5 x)}{12 y (6 y x + 5)}$

Step 5.3.3.1.5.2

Cancel the common factors.

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Step 5.3.3.1.5.2.1

Factor $12$ out of $12 y (6 y x + 5)$ .

$x' = \frac{1}{y (6 y x + 5)} - \frac{6 x^{2}}{6 y x + 5} + \frac{12 (- 5 x)}{12 (y (6 y x + 5))}$

Step 5.3.3.1.5.2.2

Cancel the common factor.

$x' = \frac{1}{y (6 y x + 5)} - \frac{6 x^{2}}{6 y x + 5} + \frac{12 (- 5 x)}{12 (y (6 y x + 5))}$

Step 5.3.3.1.5.2.3

Rewrite the expression.

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

Step 5.3.3.1.6

Move the negative in front of the fraction.

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

Step 5.3.3.2

To write $- \frac{6 x^{2}}{6 y x + 5}$ as a fraction with a common denominator, multiply by $\frac{y}{y}$ .

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

Step 5.3.3.3

Write each expression with a common denominator of $y (6 y x + 5)$ , by multiplying each by an appropriate factor of $1$ .

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Step 5.3.3.3.1

Multiply $\frac{6 x^{2}}{6 y x + 5}$ by $\frac{y}{y}$ .

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

Step 5.3.3.3.2

Reorder the factors of $(6 y x + 5) y$ .

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

Step 5.3.3.4

Combine the numerators over the common denominator.

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

Step 5.3.3.5

Combine the numerators over the common denominator.

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

Step 6

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

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