Calculus Examples

$y = \frac{3 x - 9}{2 x + 8}$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (y) = \frac{d}{d x} (\frac{3 x - 9}{2 x + 8})$

Step 2

The derivative of $y$ with respect to $x$ is $y'$ .

$y'$

Step 3

Differentiate the right side of the equation.

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Step 3.1

Differentiate using the Quotient Rule which states that $\frac{d}{d x} [\frac{f (x)}{g (x)}]$ is $\frac{g (x) \frac{d}{d x} [f (x)] - f (x) \frac{d}{d x} [g (x)]}{{g (x)}^{2}}$ where $f (x) = 3 x - 9$ and $g (x) = 2 x + 8$ .

$\frac{(2 x + 8) \frac{d}{d x} [3 x - 9] - (3 x - 9) \frac{d}{d x} [2 x + 8]}{{(2 x + 8)}^{2}}$

Step 3.2

Differentiate.

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Step 3.2.1

By the Sum Rule, the derivative of $3 x - 9$ with respect to $x$ is $\frac{d}{d x} [3 x] + \frac{d}{d x} [- 9]$ .

$\frac{(2 x + 8) (\frac{d}{d x} [3 x] + \frac{d}{d x} [- 9]) - (3 x - 9) \frac{d}{d x} [2 x + 8]}{{(2 x + 8)}^{2}}$

Step 3.2.2

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

$\frac{(2 x + 8) (3 \frac{d}{d x} [x] + \frac{d}{d x} [- 9]) - (3 x - 9) \frac{d}{d x} [2 x + 8]}{{(2 x + 8)}^{2}}$

Step 3.2.3

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

$\frac{(2 x + 8) (3 \cdot 1 + \frac{d}{d x} [- 9]) - (3 x - 9) \frac{d}{d x} [2 x + 8]}{{(2 x + 8)}^{2}}$

Step 3.2.4

Multiply $3$ by $1$ .

$\frac{(2 x + 8) (3 + \frac{d}{d x} [- 9]) - (3 x - 9) \frac{d}{d x} [2 x + 8]}{{(2 x + 8)}^{2}}$

Step 3.2.5

Since $- 9$ is constant with respect to $x$ , the derivative of $- 9$ with respect to $x$ is $0$ .

$\frac{(2 x + 8) (3 + 0) - (3 x - 9) \frac{d}{d x} [2 x + 8]}{{(2 x + 8)}^{2}}$

Step 3.2.6

Simplify the expression.

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Step 3.2.6.1

Add $3$ and $0$ .

$\frac{(2 x + 8) \cdot 3 - (3 x - 9) \frac{d}{d x} [2 x + 8]}{{(2 x + 8)}^{2}}$

Step 3.2.6.2

Move $3$ to the left of $2 x + 8$ .

$\frac{3 \cdot (2 x + 8) - (3 x - 9) \frac{d}{d x} [2 x + 8]}{{(2 x + 8)}^{2}}$

Step 3.2.7

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

$\frac{3 (2 x + 8) - (3 x - 9) (\frac{d}{d x} [2 x] + \frac{d}{d x} [8])}{{(2 x + 8)}^{2}}$

Step 3.2.8

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

$\frac{3 (2 x + 8) - (3 x - 9) (2 \frac{d}{d x} [x] + \frac{d}{d x} [8])}{{(2 x + 8)}^{2}}$

Step 3.2.9

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

$\frac{3 (2 x + 8) - (3 x - 9) (2 \cdot 1 + \frac{d}{d x} [8])}{{(2 x + 8)}^{2}}$

Step 3.2.10

Multiply $2$ by $1$ .

$\frac{3 (2 x + 8) - (3 x - 9) (2 + \frac{d}{d x} [8])}{{(2 x + 8)}^{2}}$

Step 3.2.11

Since $8$ is constant with respect to $x$ , the derivative of $8$ with respect to $x$ is $0$ .

$\frac{3 (2 x + 8) - (3 x - 9) (2 + 0)}{{(2 x + 8)}^{2}}$

Step 3.2.12

Simplify the expression.

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Step 3.2.12.1

Add $2$ and $0$ .

$\frac{3 (2 x + 8) - (3 x - 9) \cdot 2}{{(2 x + 8)}^{2}}$

Step 3.2.12.2

Multiply $2$ by $- 1$ .

$\frac{3 (2 x + 8) - 2 (3 x - 9)}{{(2 x + 8)}^{2}}$

Step 3.3

Simplify.

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Step 3.3.1

Apply the distributive property.

$\frac{3 (2 x) + 3 \cdot 8 - 2 (3 x - 9)}{{(2 x + 8)}^{2}}$

Step 3.3.2

Apply the distributive property.

$\frac{3 (2 x) + 3 \cdot 8 - 2 (3 x) - 2 \cdot - 9}{{(2 x + 8)}^{2}}$

Step 3.3.3

Simplify the numerator.

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Step 3.3.3.1

Combine the opposite terms in $3 (2 x) + 3 \cdot 8 - 2 (3 x) - 2 \cdot - 9$ .

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Step 3.3.3.1.1

Reorder the factors in the terms $3 (2 x)$ and $- 2 (3 x)$ .

$\frac{2 \cdot 3 x + 3 \cdot 8 - 2 \cdot 3 x - 2 \cdot - 9}{{(2 x + 8)}^{2}}$

Step 3.3.3.1.2

Subtract $2 \cdot 3 x$ from $2 \cdot 3 x$ .

$\frac{3 \cdot 8 + 0 - 2 \cdot - 9}{{(2 x + 8)}^{2}}$

Step 3.3.3.1.3

Add $3 \cdot 8$ and $0$ .

$\frac{3 \cdot 8 - 2 \cdot - 9}{{(2 x + 8)}^{2}}$

Step 3.3.3.2

Simplify each term.

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Step 3.3.3.2.1

Multiply $3$ by $8$ .

$\frac{24 - 2 \cdot - 9}{{(2 x + 8)}^{2}}$

Step 3.3.3.2.2

Multiply $- 2$ by $- 9$ .

$\frac{24 + 18}{{(2 x + 8)}^{2}}$

Step 3.3.3.3

Add $24$ and $18$ .

$\frac{42}{{(2 x + 8)}^{2}}$

Step 3.3.4

Simplify the denominator.

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Step 3.3.4.1

Factor $2$ out of $2 x + 8$ .

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Step 3.3.4.1.1

Factor $2$ out of $2 x$ .

$\frac{42}{{(2 (x) + 8)}^{2}}$

Step 3.3.4.1.2

Factor $2$ out of $8$ .

$\frac{42}{{(2 x + 2 \cdot 4)}^{2}}$

Step 3.3.4.1.3

Factor $2$ out of $2 x + 2 \cdot 4$ .

$\frac{42}{{(2 (x + 4))}^{2}}$

Step 3.3.4.2

Apply the product rule to $2 (x + 4)$ .

$\frac{42}{2^{2} {(x + 4)}^{2}}$

Step 3.3.4.3

Raise $2$ to the power of $2$ .

$\frac{42}{4 {(x + 4)}^{2}}$

Step 3.3.5

Cancel the common factor of $42$ and $4$ .

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Step 3.3.5.1

Factor $2$ out of $42$ .

$\frac{2 (21)}{4 {(x + 4)}^{2}}$

Step 3.3.5.2

Cancel the common factors.

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Step 3.3.5.2.1

Factor $2$ out of $4 {(x + 4)}^{2}$ .

$\frac{2 (21)}{2 (2 {(x + 4)}^{2})}$

Step 3.3.5.2.2

Cancel the common factor.

$\frac{2 \cdot 21}{2 (2 {(x + 4)}^{2})}$

Step 3.3.5.2.3

Rewrite the expression.

$\frac{21}{2 {(x + 4)}^{2}}$

Step 4

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

$y' = \frac{21}{2 {(x + 4)}^{2}}$

Step 5

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

$\frac{d y}{d x} = \frac{21}{2 {(x + 4)}^{2}}$