Calculus Examples

${(2 x + 9 y)}^{2} = - 8 x^{2} - 10 y^{2}$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} ({(2 x + 9 y)}^{2}) = \frac{d}{d x} (- 8 x^{2} - 10 y^{2})$

Step 2

Differentiate the left side of the equation.

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Step 2.1

Rewrite ${(2 x + 9 y)}^{2}$ as $(2 x + 9 y) (2 x + 9 y)$ .

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

Step 2.2

Expand $(2 x + 9 y) (2 x + 9 y)$ using the FOIL Method.

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Step 2.2.1

Apply the distributive property.

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

Step 2.2.2

Apply the distributive property.

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

Step 2.2.3

Apply the distributive property.

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

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

Rewrite using the commutative property of multiplication.

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

Step 2.3.1.2

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

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Step 2.3.1.2.1

Move $x$ .

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

Step 2.3.1.2.2

Multiply $x$ by $x$ .

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

Step 2.3.1.3

Multiply $2$ by $2$ .

$\frac{d}{d x} [4 x^{2} + 2 x (9 y) + 9 y (2 x) + 9 y (9 y)]$

Step 2.3.1.4

Rewrite using the commutative property of multiplication.

$\frac{d}{d x} [4 x^{2} + 2 \cdot 9 x y + 9 y (2 x) + 9 y (9 y)]$

Step 2.3.1.5

Multiply $2$ by $9$ .

$\frac{d}{d x} [4 x^{2} + 18 x y + 9 y (2 x) + 9 y (9 y)]$

Step 2.3.1.6

Rewrite using the commutative property of multiplication.

$\frac{d}{d x} [4 x^{2} + 18 x y + 9 \cdot 2 y x + 9 y (9 y)]$

Step 2.3.1.7

Multiply $9$ by $2$ .

$\frac{d}{d x} [4 x^{2} + 18 x y + 18 y x + 9 y (9 y)]$

Step 2.3.1.8

Rewrite using the commutative property of multiplication.

$\frac{d}{d x} [4 x^{2} + 18 x y + 18 y x + 9 \cdot 9 y \cdot y]$

Step 2.3.1.9

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

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Step 2.3.1.9.1

Move $y$ .

$\frac{d}{d x} [4 x^{2} + 18 x y + 18 y x + 9 \cdot 9 (y \cdot y)]$

Step 2.3.1.9.2

Multiply $y$ by $y$ .

$\frac{d}{d x} [4 x^{2} + 18 x y + 18 y x + 9 \cdot 9 y^{2}]$

Step 2.3.1.10

Multiply $9$ by $9$ .

$\frac{d}{d x} [4 x^{2} + 18 x y + 18 y x + 81 y^{2}]$

Step 2.3.2

Add $18 x y$ and $18 y x$ .

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Step 2.3.2.1

Move $y$ .

$\frac{d}{d x} [4 x^{2} + 18 x y + 18 x y + 81 y^{2}]$

Step 2.3.2.2

Add $18 x y$ and $18 x y$ .

$\frac{d}{d x} [4 x^{2} + 36 x y + 81 y^{2}]$

Step 2.4

Differentiate.

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Step 2.4.1

By the Sum Rule, the derivative of $4 x^{2} + 36 x y + 81 y^{2}$ with respect to $x$ is $\frac{d}{d x} [4 x^{2}] + \frac{d}{d x} [36 x y] + \frac{d}{d x} [81 y^{2}]$ .

$\frac{d}{d x} [4 x^{2}] + \frac{d}{d x} [36 x y] + \frac{d}{d x} [81 y^{2}]$

Step 2.4.2

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

$4 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [36 x y] + \frac{d}{d x} [81 y^{2}]$

Step 2.4.3

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

$4 (2 x) + \frac{d}{d x} [36 x y] + \frac{d}{d x} [81 y^{2}]$

Step 2.4.4

Multiply $2$ by $4$ .

$8 x + \frac{d}{d x} [36 x y] + \frac{d}{d x} [81 y^{2}]$

Step 2.4.5

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

$8 x + 36 \frac{d}{d x} [x y] + \frac{d}{d x} [81 y^{2}]$

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

$8 x + 36 (x \frac{d}{d x} [y] + y \frac{d}{d x} [x]) + \frac{d}{d x} [81 y^{2}]$

Step 2.6

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

$8 x + 36 (x y' + y \frac{d}{d x} [x]) + \frac{d}{d x} [81 y^{2}]$

Step 2.7

Differentiate.

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Step 2.7.1

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

$8 x + 36 (x y' + y \cdot 1) + \frac{d}{d x} [81 y^{2}]$

Step 2.7.2

Multiply $y$ by $1$ .

$8 x + 36 (x y' + y) + \frac{d}{d x} [81 y^{2}]$

Step 2.7.3

Since $81$ is constant with respect to $x$ , the derivative of $81 y^{2}$ with respect to $x$ is $81 \frac{d}{d x} [y^{2}]$ .

$8 x + 36 (x y' + y) + 81 \frac{d}{d x} [y^{2}]$

Step 2.8

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 2.8.1

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

$8 x + 36 (x y' + y) + 81 (\frac{d}{d u} [u^{2}] \frac{d}{d x} [y])$

Step 2.8.2

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

$8 x + 36 (x y' + y) + 81 (2 u \frac{d}{d x} [y])$

Step 2.8.3

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

$8 x + 36 (x y' + y) + 81 (2 y \frac{d}{d x} [y])$

Step 2.9

Multiply $2$ by $81$ .

$8 x + 36 (x y' + y) + 162 (y \frac{d}{d x} [y])$

Step 2.10

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

$8 x + 36 (x y' + y) + 162 y y'$

Step 2.11

Simplify.

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Step 2.11.1

Apply the distributive property.

$8 x + 36 (x y') + 36 y + 162 y y'$

Step 2.11.2

Remove unnecessary parentheses.

$8 x + 36 x y' + 36 y + 162 y y'$

Step 2.11.3

Reorder terms.

$36 x y' + 162 y y' + 8 x + 36 y$

Step 3

Differentiate the right side of the equation.

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Step 3.1

By the Sum Rule, the derivative of $- 8 x^{2} - 10 y^{2}$ with respect to $x$ is $\frac{d}{d x} [- 8 x^{2}] + \frac{d}{d x} [- 10 y^{2}]$ .

$\frac{d}{d x} [- 8 x^{2}] + \frac{d}{d x} [- 10 y^{2}]$

Step 3.2

Evaluate $\frac{d}{d x} [- 8 x^{2}]$ .

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Step 3.2.1

Since $- 8$ is constant with respect to $x$ , the derivative of $- 8 x^{2}$ with respect to $x$ is $- 8 \frac{d}{d x} [x^{2}]$ .

$- 8 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 10 y^{2}]$

Step 3.2.2

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

$- 8 (2 x) + \frac{d}{d x} [- 10 y^{2}]$

Step 3.2.3

Multiply $2$ by $- 8$ .

$- 16 x + \frac{d}{d x} [- 10 y^{2}]$

Step 3.3

Evaluate $\frac{d}{d x} [- 10 y^{2}]$ .

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Step 3.3.1

Since $- 10$ is constant with respect to $x$ , the derivative of $- 10 y^{2}$ with respect to $x$ is $- 10 \frac{d}{d x} [y^{2}]$ .

$- 16 x - 10 \frac{d}{d x} [y^{2}]$

Step 3.3.2

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

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Step 3.3.2.1

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

$- 16 x - 10 (\frac{d}{d u} [u^{2}] \frac{d}{d x} [y])$

Step 3.3.2.2

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

$- 16 x - 10 (2 u \frac{d}{d x} [y])$

Step 3.3.2.3

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

$- 16 x - 10 (2 y \frac{d}{d x} [y])$

Step 3.3.3

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

$- 16 x - 10 (2 y y')$

Step 3.3.4

Multiply $2$ by $- 10$ .

$- 16 x - 20 y y'$

Step 3.4

Reorder terms.

$- 20 y y' - 16 x$

Step 4

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

$36 x y' + 162 y y' + 8 x + 36 y = - 20 y y' - 16 x$

Step 5

Solve for $y'$ .

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Step 5.1

Move all terms containing $y'$ to the left side of the equation.

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Step 5.1.1

Add $20 y y'$ to both sides of the equation.

$36 x y' + 162 y y' + 8 x + 36 y + 20 y y' = - 16 x$

Step 5.1.2

Add $162 y y'$ and $20 y y'$ .

$36 x y' + 182 y y' + 8 x + 36 y = - 16 x$

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 $8 x$ from both sides of the equation.

$36 x y' + 182 y y' + 36 y = - 16 x - 8 x$

Step 5.2.2

Subtract $36 y$ from both sides of the equation.

$36 x y' + 182 y y' = - 16 x - 8 x - 36 y$

Step 5.2.3

Subtract $8 x$ from $- 16 x$ .

$36 x y' + 182 y y' = - 24 x - 36 y$

Step 5.3

Factor $2 y'$ out of $36 x y' + 182 y y'$ .

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Step 5.3.1

Factor $2 y'$ out of $36 x y'$ .

$2 y' (18 x) + 182 y y' = - 24 x - 36 y$

Step 5.3.2

Factor $2 y'$ out of $182 y y'$ .

$2 y' (18 x) + 2 y' (91 y) = - 24 x - 36 y$

Step 5.3.3

Factor $2 y'$ out of $2 y' (18 x) + 2 y' (91 y)$ .

$2 y' (18 x + 91 y) = - 24 x - 36 y$

Step 5.4

Divide each term in $2 y' (18 x + 91 y) = - 24 x - 36 y$ by $2 (18 x + 91 y)$ and simplify.

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Step 5.4.1

Divide each term in $2 y' (18 x + 91 y) = - 24 x - 36 y$ by $2 (18 x + 91 y)$ .

$\frac{2 y' (18 x + 91 y)}{2 (18 x + 91 y)} = \frac{- 24 x}{2 (18 x + 91 y)} + \frac{- 36 y}{2 (18 x + 91 y)}$

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.

$\frac{2 y' (18 x + 91 y)}{2 (18 x + 91 y)} = \frac{- 24 x}{2 (18 x + 91 y)} + \frac{- 36 y}{2 (18 x + 91 y)}$

Step 5.4.2.1.2

Rewrite the expression.

$\frac{y' (18 x + 91 y)}{18 x + 91 y} = \frac{- 24 x}{2 (18 x + 91 y)} + \frac{- 36 y}{2 (18 x + 91 y)}$

Step 5.4.2.2

Cancel the common factor of $18 x + 91 y$ .

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Step 5.4.2.2.1

Cancel the common factor.

$\frac{y' (18 x + 91 y)}{18 x + 91 y} = \frac{- 24 x}{2 (18 x + 91 y)} + \frac{- 36 y}{2 (18 x + 91 y)}$

Step 5.4.2.2.2

Divide $y'$ by $1$ .

$y' = \frac{- 24 x}{2 (18 x + 91 y)} + \frac{- 36 y}{2 (18 x + 91 y)}$

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 $- 24$ and $2$ .

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Step 5.4.3.1.1.1

Factor $2$ out of $- 24 x$ .

$y' = \frac{2 (- 12 x)}{2 (18 x + 91 y)} + \frac{- 36 y}{2 (18 x + 91 y)}$

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.

$y' = \frac{2 (- 12 x)}{2 (18 x + 91 y)} + \frac{- 36 y}{2 (18 x + 91 y)}$

Step 5.4.3.1.1.2.2

Rewrite the expression.

$y' = \frac{- 12 x}{18 x + 91 y} + \frac{- 36 y}{2 (18 x + 91 y)}$

Step 5.4.3.1.2

Move the negative in front of the fraction.

$y' = - \frac{12 x}{18 x + 91 y} + \frac{- 36 y}{2 (18 x + 91 y)}$

Step 5.4.3.1.3

Cancel the common factor of $- 36$ and $2$ .

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Step 5.4.3.1.3.1

Factor $2$ out of $- 36 y$ .

$y' = - \frac{12 x}{18 x + 91 y} + \frac{2 (- 18 y)}{2 (18 x + 91 y)}$

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.

$y' = - \frac{12 x}{18 x + 91 y} + \frac{2 (- 18 y)}{2 (18 x + 91 y)}$

Step 5.4.3.1.3.2.2

Rewrite the expression.

$y' = - \frac{12 x}{18 x + 91 y} + \frac{- 18 y}{18 x + 91 y}$

Step 5.4.3.1.4

Move the negative in front of the fraction.

$y' = - \frac{12 x}{18 x + 91 y} - \frac{18 y}{18 x + 91 y}$

Step 5.4.3.2

Simplify terms.

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Step 5.4.3.2.1

Combine the numerators over the common denominator.

$y' = \frac{- 12 x - 18 y}{18 x + 91 y}$

Step 5.4.3.2.2

Factor $6$ out of $- 12 x - 18 y$ .

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Step 5.4.3.2.2.1

Factor $6$ out of $- 12 x$ .

$y' = \frac{6 (- 2 x) - 18 y}{18 x + 91 y}$

Step 5.4.3.2.2.2

Factor $6$ out of $- 18 y$ .

$y' = \frac{6 (- 2 x) + 6 (- 3 y)}{18 x + 91 y}$

Step 5.4.3.2.2.3

Factor $6$ out of $6 (- 2 x) + 6 (- 3 y)$ .

$y' = \frac{6 (- 2 x - 3 y)}{18 x + 91 y}$

Step 5.4.3.2.3

Factor $- 1$ out of $- 2 x$ .

$y' = \frac{6 (- (2 x) - 3 y)}{18 x + 91 y}$

Step 5.4.3.2.4

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

$y' = \frac{6 (- (2 x) - (3 y))}{18 x + 91 y}$

Step 5.4.3.2.5

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

$y' = \frac{6 (- (2 x + 3 y))}{18 x + 91 y}$

Step 5.4.3.2.6

Simplify the expression.

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Step 5.4.3.2.6.1

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

$y' = \frac{6 (- 1 (2 x + 3 y))}{18 x + 91 y}$

Step 5.4.3.2.6.2

Move the negative in front of the fraction.

$y' = - \frac{6 (2 x + 3 y)}{18 x + 91 y}$

Step 6

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

$\frac{d y}{d x} = - \frac{6 (2 x + 3 y)}{18 x + 91 y}$