Calculus Examples

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

Step 1

Differentiate both sides of the equation.

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

Step 2

Differentiate the left side of the equation.

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Step 2.1

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

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

Step 2.2

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

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Step 2.2.1

Apply the distributive property.

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

Step 2.2.2

Apply the distributive property.

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

Step 2.2.3

Apply the distributive property.

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

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.

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

Step 2.3.1.2

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

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Step 2.3.1.2.1

Move $x$ .

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

Step 2.3.1.2.2

Multiply $x$ by $x$ .

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

Step 2.3.1.3

Multiply $6$ by $6$ .

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

Step 2.3.1.4

Rewrite using the commutative property of multiplication.

$\frac{d}{d x} [36 x^{2} + 6 \cdot - 1 x y - y (6 x) - y (- y) - 5 y]$

Step 2.3.1.5

Multiply $6$ by $- 1$ .

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

Step 2.3.1.6

Rewrite using the commutative property of multiplication.

$\frac{d}{d x} [36 x^{2} - 6 x y - 1 \cdot 6 y x - y (- y) - 5 y]$

Step 2.3.1.7

Multiply $- 1$ by $6$ .

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

Step 2.3.1.8

Rewrite using the commutative property of multiplication.

$\frac{d}{d x} [36 x^{2} - 6 x y - 6 y x - 1 \cdot - 1 y \cdot y - 5 y]$

Step 2.3.1.9

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

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Step 2.3.1.9.1

Move $y$ .

$\frac{d}{d x} [36 x^{2} - 6 x y - 6 y x - 1 \cdot - 1 (y \cdot y) - 5 y]$

Step 2.3.1.9.2

Multiply $y$ by $y$ .

$\frac{d}{d x} [36 x^{2} - 6 x y - 6 y x - 1 \cdot - 1 y^{2} - 5 y]$

Step 2.3.1.10

Multiply $- 1$ by $- 1$ .

$\frac{d}{d x} [36 x^{2} - 6 x y - 6 y x + 1 y^{2} - 5 y]$

Step 2.3.1.11

Multiply $y^{2}$ by $1$ .

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

Step 2.3.2

Subtract $6 y x$ from $- 6 x y$ .

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Step 2.3.2.1

Move $y$ .

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

Step 2.3.2.2

Subtract $6 x y$ from $- 6 x y$ .

$\frac{d}{d x} [36 x^{2} - 12 x y + y^{2} - 5 y]$

Step 2.4

By the Sum Rule, the derivative of $36 x^{2} - 12 x y + y^{2} - 5 y$ with respect to $x$ is $\frac{d}{d x} [36 x^{2}] + \frac{d}{d x} [- 12 x y] + \frac{d}{d x} [y^{2}] + \frac{d}{d x} [- 5 y]$ .

$\frac{d}{d x} [36 x^{2}] + \frac{d}{d x} [- 12 x y] + \frac{d}{d x} [y^{2}] + \frac{d}{d x} [- 5 y]$

Step 2.5

Evaluate $\frac{d}{d x} [36 x^{2}]$ .

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Step 2.5.1

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

$36 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 12 x y] + \frac{d}{d x} [y^{2}] + \frac{d}{d x} [- 5 y]$

Step 2.5.2

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

$36 (2 x) + \frac{d}{d x} [- 12 x y] + \frac{d}{d x} [y^{2}] + \frac{d}{d x} [- 5 y]$

Step 2.5.3

Multiply $2$ by $36$ .

$72 x + \frac{d}{d x} [- 12 x y] + \frac{d}{d x} [y^{2}] + \frac{d}{d x} [- 5 y]$

Step 2.6

Evaluate $\frac{d}{d x} [- 12 x y]$ .

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Step 2.6.1

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

$72 x - 12 \frac{d}{d x} [x y] + \frac{d}{d x} [y^{2}] + \frac{d}{d x} [- 5 y]$

Step 2.6.2

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 $g (x) = y$ .

$72 x - 12 (x \frac{d}{d x} [y] + y \frac{d}{d x} [x]) + \frac{d}{d x} [y^{2}] + \frac{d}{d x} [- 5 y]$

Step 2.6.3

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

$72 x - 12 (x y' + y \frac{d}{d x} [x]) + \frac{d}{d x} [y^{2}] + \frac{d}{d x} [- 5 y]$

Step 2.6.4

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

$72 x - 12 (x y' + y \cdot 1) + \frac{d}{d x} [y^{2}] + \frac{d}{d x} [- 5 y]$

Step 2.6.5

Multiply $y$ by $1$ .

$72 x - 12 (x y' + y) + \frac{d}{d x} [y^{2}] + \frac{d}{d x} [- 5 y]$

Step 2.7

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

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Step 2.7.1

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 2.7.1.1

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

$72 x - 12 (x y' + y) + \frac{d}{d u} [u^{2}] \frac{d}{d x} [y] + \frac{d}{d x} [- 5 y]$

Step 2.7.1.2

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

$72 x - 12 (x y' + y) + 2 u \frac{d}{d x} [y] + \frac{d}{d x} [- 5 y]$

Step 2.7.1.3

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

$72 x - 12 (x y' + y) + 2 y \frac{d}{d x} [y] + \frac{d}{d x} [- 5 y]$

Step 2.7.2

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

$72 x - 12 (x y' + y) + 2 y y' + \frac{d}{d x} [- 5 y]$

Step 2.8

Evaluate $\frac{d}{d x} [- 5 y]$ .

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Step 2.8.1

Since $- 5$ is constant with respect to $x$ , the derivative of $- 5 y$ with respect to $x$ is $- 5 \frac{d}{d x} [y]$ .

$72 x - 12 (x y' + y) + 2 y y' - 5 \frac{d}{d x} [y]$

Step 2.8.2

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

$72 x - 12 (x y' + y) + 2 y y' - 5 y'$

Step 2.9

Simplify.

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Step 2.9.1

Apply the distributive property.

$72 x - 12 (x y') - 12 y + 2 y y' - 5 y'$

Step 2.9.2

Remove unnecessary parentheses.

$72 x - 12 x y' - 12 y + 2 y y' - 5 y'$

Step 2.9.3

Reorder terms.

$- 12 x y' + 2 y y' + 72 x - 12 y - 5 y'$

Step 3

Since $2$ is constant with respect to $x$ , the derivative of $2$ with respect to $x$ is $0$ .

$0$

Step 4

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

$- 12 x y' + 2 y y' + 72 x - 12 y - 5 y' = 0$

Step 5

Solve for $y'$ .

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Step 5.1

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

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Step 5.1.1

Subtract $72 x$ from both sides of the equation.

$- 12 x y' + 2 y y' - 12 y - 5 y' = - 72 x$

Step 5.1.2

Add $12 y$ to both sides of the equation.

$- 12 x y' + 2 y y' - 5 y' = - 72 x + 12 y$

Step 5.2

Factor $y'$ out of $- 12 x y' + 2 y y' - 5 y'$ .

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Step 5.2.1

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

$y' (- 12 x) + 2 y y' - 5 y' = - 72 x + 12 y$

Step 5.2.2

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

$y' (- 12 x) + y' (2 y) - 5 y' = - 72 x + 12 y$

Step 5.2.3

Factor $y'$ out of $- 5 y'$ .

$y' (- 12 x) + y' (2 y) + y' \cdot - 5 = - 72 x + 12 y$

Step 5.2.4

Factor $y'$ out of $y' (- 12 x) + y' (2 y)$ .

$y' (- 12 x + 2 y) + y' \cdot - 5 = - 72 x + 12 y$

Step 5.2.5

Factor $y'$ out of $y' (- 12 x + 2 y) + y' \cdot - 5$ .

$y' (- 12 x + 2 y - 5) = - 72 x + 12 y$

Step 5.3

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

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Step 5.3.1

Divide each term in $y' (- 12 x + 2 y - 5) = - 72 x + 12 y$ by $- 12 x + 2 y - 5$ .

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

Step 5.3.2

Simplify the left side.

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Step 5.3.2.1

Cancel the common factor of $- 12 x + 2 y - 5$ .

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Step 5.3.2.1.1

Cancel the common factor.

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

Step 5.3.2.1.2

Divide $y'$ by $1$ .

$y' = \frac{- 72 x}{- 12 x + 2 y - 5} + \frac{12 y}{- 12 x + 2 y - 5}$

Step 5.3.3

Simplify the right side.

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Step 5.3.3.1

Combine the numerators over the common denominator.

$y' = \frac{- 72 x + 12 y}{- 12 x + 2 y - 5}$

Step 5.3.3.2

Factor $12$ out of $- 72 x + 12 y$ .

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Step 5.3.3.2.1

Factor $12$ out of $- 72 x$ .

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

Step 5.3.3.2.2

Factor $12$ out of $12 y$ .

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

Step 5.3.3.2.3

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

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

Step 5.3.3.3

Factor $- 1$ out of $- 6 x$ .

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

Step 5.3.3.4

Factor $- 1$ out of $y$ .

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

Step 5.3.3.5

Factor $- 1$ out of $- (6 x) - 1 (- y)$ .

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

Step 5.3.3.6

Rewrite $- (6 x - y)$ as $- 1 (6 x - y)$ .

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

Step 5.3.3.7

Factor $- 1$ out of $- 12 x$ .

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

Step 5.3.3.8

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

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

Step 5.3.3.9

Factor $- 1$ out of $- (12 x) - (- 2 y)$ .

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

Step 5.3.3.10

Rewrite $- 5$ as $- 1 (5)$ .

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

Step 5.3.3.11

Factor $- 1$ out of $- (12 x - 2 y) - 1 (5)$ .

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

Step 5.3.3.12

Rewrite $- (12 x - 2 y + 5)$ as $- 1 (12 x - 2 y + 5)$ .

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

Step 5.3.3.13

Cancel the common factor.

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

Step 5.3.3.14

Rewrite the expression.

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

Step 6

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

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