Precalculus Examples

$y = \frac{3 x + 2}{4 x - 7}$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d y} (y) = \frac{d}{d y} (\frac{3 x + 2}{4 x - 7})$

Step 2

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

$1$

Step 3

Differentiate the right side of the equation.

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Step 3.1

Differentiate using the Quotient Rule which states that $\frac{d}{d y} [\frac{f (y)}{g (y)}]$ is $\frac{g (y) \frac{d}{d y} [f (y)] - f (y) \frac{d}{d y} [g (y)]}{{g (y)}^{2}}$ where $f (y) = 3 x + 2$ and $g (y) = 4 x - 7$ .

$\frac{(4 x - 7) \frac{d}{d y} [3 x + 2] - (3 x + 2) \frac{d}{d y} [4 x - 7]}{{(4 x - 7)}^{2}}$

Step 3.2

Differentiate.

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Step 3.2.1

By the Sum Rule, the derivative of $3 x + 2$ with respect to $y$ is $\frac{d}{d y} [3 x] + \frac{d}{d y} [2]$ .

$\frac{(4 x - 7) (\frac{d}{d y} [3 x] + \frac{d}{d y} [2]) - (3 x + 2) \frac{d}{d y} [4 x - 7]}{{(4 x - 7)}^{2}}$

Step 3.2.2

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

$\frac{(4 x - 7) (3 \frac{d}{d y} [x] + \frac{d}{d y} [2]) - (3 x + 2) \frac{d}{d y} [4 x - 7]}{{(4 x - 7)}^{2}}$

Step 3.3

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

$\frac{(4 x - 7) (3 x' + \frac{d}{d y} [2]) - (3 x + 2) \frac{d}{d y} [4 x - 7]}{{(4 x - 7)}^{2}}$

Step 3.4

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

$\frac{(4 x - 7) (3 x' + 0) - (3 x + 2) \frac{d}{d y} [4 x - 7]}{{(4 x - 7)}^{2}}$

Step 3.5

Simplify the expression.

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Step 3.5.1

Add $3 x'$ and $0$ .

$\frac{(4 x - 7) (3 x') - (3 x + 2) \frac{d}{d y} [4 x - 7]}{{(4 x - 7)}^{2}}$

Step 3.5.2

Move $3$ to the left of $4 x - 7$ .

$\frac{3 \cdot (4 x - 7) x' - (3 x + 2) \frac{d}{d y} [4 x - 7]}{{(4 x - 7)}^{2}}$

Step 3.6

By the Sum Rule, the derivative of $4 x - 7$ with respect to $y$ is $\frac{d}{d y} [4 x] + \frac{d}{d y} [- 7]$ .

$\frac{3 (4 x - 7) x' - (3 x + 2) (\frac{d}{d y} [4 x] + \frac{d}{d y} [- 7])}{{(4 x - 7)}^{2}}$

Step 3.7

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

$\frac{3 (4 x - 7) x' - (3 x + 2) (4 \frac{d}{d y} [x] + \frac{d}{d y} [- 7])}{{(4 x - 7)}^{2}}$

Step 3.8

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

$\frac{3 (4 x - 7) x' - (3 x + 2) (4 x' + \frac{d}{d y} [- 7])}{{(4 x - 7)}^{2}}$

Step 3.9

Since $- 7$ is constant with respect to $y$ , the derivative of $- 7$ with respect to $y$ is $0$ .

$\frac{3 (4 x - 7) x' - (3 x + 2) (4 x' + 0)}{{(4 x - 7)}^{2}}$

Step 3.10

Simplify the expression.

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Step 3.10.1

Add $4 x'$ and $0$ .

$\frac{3 (4 x - 7) x' - (3 x + 2) (4 x')}{{(4 x - 7)}^{2}}$

Step 3.10.2

Multiply $4$ by $- 1$ .

$\frac{3 (4 x - 7) x' - 4 (3 x + 2) x'}{{(4 x - 7)}^{2}}$

Step 3.11

Simplify.

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Step 3.11.1

Apply the distributive property.

$\frac{(3 (4 x) + 3 \cdot - 7) x' - 4 (3 x + 2) x'}{{(4 x - 7)}^{2}}$

Step 3.11.2

Apply the distributive property.

$\frac{3 (4 x) x' + 3 \cdot - 7 x' - 4 (3 x + 2) x'}{{(4 x - 7)}^{2}}$

Step 3.11.3

Apply the distributive property.

$\frac{3 (4 x) x' + 3 \cdot - 7 x' + (- 4 (3 x) - 4 \cdot 2) x'}{{(4 x - 7)}^{2}}$

Step 3.11.4

Apply the distributive property.

$\frac{3 (4 x) x' + 3 \cdot - 7 x' - 4 (3 x) x' - 4 \cdot 2 x'}{{(4 x - 7)}^{2}}$

Step 3.11.5

Simplify the numerator.

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Step 3.11.5.1

Simplify each term.

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Step 3.11.5.1.1

Multiply $4$ by $3$ .

$\frac{12 x x' + 3 \cdot - 7 x' - 4 (3 x) x' - 4 \cdot 2 x'}{{(4 x - 7)}^{2}}$

Step 3.11.5.1.2

Multiply $3$ by $- 7$ .

$\frac{12 x x' - 21 x' - 4 (3 x) x' - 4 \cdot 2 x'}{{(4 x - 7)}^{2}}$

Step 3.11.5.1.3

Multiply $3$ by $- 4$ .

$\frac{12 x x' - 21 x' - 12 x x' - 4 \cdot 2 x'}{{(4 x - 7)}^{2}}$

Step 3.11.5.1.4

Multiply $- 4$ by $2$ .

$\frac{12 x x' - 21 x' - 12 x x' - 8 x'}{{(4 x - 7)}^{2}}$

Step 3.11.5.2

Combine the opposite terms in $12 x x' - 21 x' - 12 x x' - 8 x'$ .

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Step 3.11.5.2.1

Subtract $12 x x'$ from $12 x x'$ .

$\frac{0 - 21 x' - 8 x'}{{(4 x - 7)}^{2}}$

Step 3.11.5.2.2

Subtract $21 x'$ from $0$ .

$\frac{- 21 x' - 8 x'}{{(4 x - 7)}^{2}}$

Step 3.11.5.3

Subtract $8 x'$ from $- 21 x'$ .

$\frac{- 29 x'}{{(4 x - 7)}^{2}}$

Step 3.11.6

Move the negative in front of the fraction.

$- \frac{29 x'}{{(4 x - 7)}^{2}}$

Step 4

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

$1 = - \frac{29 x'}{{(4 x - 7)}^{2}}$

Step 5

Solve for $x'$ .

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Step 5.1

Rewrite the equation as $- \frac{29 x'}{{(4 x - 7)}^{2}} = 1$ .

$- \frac{29 x'}{{(4 x - 7)}^{2}} = 1$

Step 5.2

Divide each term in $- \frac{29 x'}{{(4 x - 7)}^{2}} = 1$ by $- 1$ and simplify.

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Step 5.2.1

Divide each term in $- \frac{29 x'}{{(4 x - 7)}^{2}} = 1$ by $- 1$ .

$\frac{- \frac{29 x'}{{(4 x - 7)}^{2}}}{- 1} = \frac{1}{- 1}$

Step 5.2.2

Simplify the left side.

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Step 5.2.2.1

Dividing two negative values results in a positive value.

$\frac{\frac{29 x'}{{(4 x - 7)}^{2}}}{1} = \frac{1}{- 1}$

Step 5.2.2.2

Divide $\frac{29 x'}{{(4 x - 7)}^{2}}$ by $1$ .

$\frac{29 x'}{{(4 x - 7)}^{2}} = \frac{1}{- 1}$

Step 5.2.3

Simplify the right side.

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Step 5.2.3.1

Divide $1$ by $- 1$ .

$\frac{29 x'}{{(4 x - 7)}^{2}} = - 1$

Step 5.3

Multiply both sides by ${(4 x - 7)}^{2}$ .

$\frac{29 x'}{{(4 x - 7)}^{2}} {(4 x - 7)}^{2} = - {(4 x - 7)}^{2}$

Step 5.4

Simplify the left side.

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Step 5.4.1

Cancel the common factor of ${(4 x - 7)}^{2}$ .

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Step 5.4.1.1

Cancel the common factor.

$\frac{29 x'}{{(4 x - 7)}^{2}} {(4 x - 7)}^{2} = - {(4 x - 7)}^{2}$

Step 5.4.1.2

Rewrite the expression.

$29 x' = - {(4 x - 7)}^{2}$

Step 5.5

Divide each term in $29 x' = - {(4 x - 7)}^{2}$ by $29$ and simplify.

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Step 5.5.1

Divide each term in $29 x' = - {(4 x - 7)}^{2}$ by $29$ .

$\frac{29 x'}{29} = \frac{- {(4 x - 7)}^{2}}{29}$

Step 5.5.2

Simplify the left side.

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Step 5.5.2.1

Cancel the common factor of $29$ .

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Step 5.5.2.1.1

Cancel the common factor.

$\frac{29 x'}{29} = \frac{- {(4 x - 7)}^{2}}{29}$

Step 5.5.2.1.2

Divide $x'$ by $1$ .

$x' = \frac{- {(4 x - 7)}^{2}}{29}$

Step 5.5.3

Simplify the right side.

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Step 5.5.3.1

Move the negative in front of the fraction.

$x' = - \frac{{(4 x - 7)}^{2}}{29}$

Step 6

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

$\frac{d x}{d y} = - \frac{{(4 x - 7)}^{2}}{29}$