Calculus Examples

$\frac{3 x - 7}{2 x + 9}$

Step 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 $f (x) = 3 x - 7$ and $g (x) = 2 x + 9$ .

$\frac{(2 x + 9) \frac{d}{d x} [3 x - 7] - (3 x - 7) \frac{d}{d x} [2 x + 9]}{{(2 x + 9)}^{2}}$

Step 2

Differentiate.

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Step 2.1

By the Sum Rule, the derivative of $3 x - 7$ with respect to $x$ is $\frac{d}{d x} [3 x] + \frac{d}{d x} [- 7]$ .

$\frac{(2 x + 9) (\frac{d}{d x} [3 x] + \frac{d}{d x} [- 7]) - (3 x - 7) \frac{d}{d x} [2 x + 9]}{{(2 x + 9)}^{2}}$

Step 2.2

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

$\frac{(2 x + 9) (3 \frac{d}{d x} [x] + \frac{d}{d x} [- 7]) - (3 x - 7) \frac{d}{d x} [2 x + 9]}{{(2 x + 9)}^{2}}$

Step 2.3

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

$\frac{(2 x + 9) (3 \cdot 1 + \frac{d}{d x} [- 7]) - (3 x - 7) \frac{d}{d x} [2 x + 9]}{{(2 x + 9)}^{2}}$

Step 2.4

Multiply $3$ by $1$ .

$\frac{(2 x + 9) (3 + \frac{d}{d x} [- 7]) - (3 x - 7) \frac{d}{d x} [2 x + 9]}{{(2 x + 9)}^{2}}$

Step 2.5

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

$\frac{(2 x + 9) (3 + 0) - (3 x - 7) \frac{d}{d x} [2 x + 9]}{{(2 x + 9)}^{2}}$

Step 2.6

Simplify the expression.

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Step 2.6.1

Add $3$ and $0$ .

$\frac{(2 x + 9) \cdot 3 - (3 x - 7) \frac{d}{d x} [2 x + 9]}{{(2 x + 9)}^{2}}$

Step 2.6.2

Move $3$ to the left of $2 x + 9$ .

$\frac{3 \cdot (2 x + 9) - (3 x - 7) \frac{d}{d x} [2 x + 9]}{{(2 x + 9)}^{2}}$

Step 2.7

By the Sum Rule, the derivative of $2 x + 9$ with respect to $x$ is $\frac{d}{d x} [2 x] + \frac{d}{d x} [9]$ .

$\frac{3 (2 x + 9) - (3 x - 7) (\frac{d}{d x} [2 x] + \frac{d}{d x} [9])}{{(2 x + 9)}^{2}}$

Step 2.8

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

$\frac{3 (2 x + 9) - (3 x - 7) (2 \frac{d}{d x} [x] + \frac{d}{d x} [9])}{{(2 x + 9)}^{2}}$

Step 2.9

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

$\frac{3 (2 x + 9) - (3 x - 7) (2 \cdot 1 + \frac{d}{d x} [9])}{{(2 x + 9)}^{2}}$

Step 2.10

Multiply $2$ by $1$ .

$\frac{3 (2 x + 9) - (3 x - 7) (2 + \frac{d}{d x} [9])}{{(2 x + 9)}^{2}}$

Step 2.11

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

$\frac{3 (2 x + 9) - (3 x - 7) (2 + 0)}{{(2 x + 9)}^{2}}$

Step 2.12

Simplify the expression.

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Step 2.12.1

Add $2$ and $0$ .

$\frac{3 (2 x + 9) - (3 x - 7) \cdot 2}{{(2 x + 9)}^{2}}$

Step 2.12.2

Multiply $2$ by $- 1$ .

$\frac{3 (2 x + 9) - 2 (3 x - 7)}{{(2 x + 9)}^{2}}$

Step 3

Simplify.

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Step 3.1

Apply the distributive property.

$\frac{3 (2 x) + 3 \cdot 9 - 2 (3 x - 7)}{{(2 x + 9)}^{2}}$

Step 3.2

Apply the distributive property.

$\frac{3 (2 x) + 3 \cdot 9 - 2 (3 x) - 2 \cdot - 7}{{(2 x + 9)}^{2}}$

Step 3.3

Simplify the numerator.

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Step 3.3.1

Combine the opposite terms in $3 (2 x) + 3 \cdot 9 - 2 (3 x) - 2 \cdot - 7$ .

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Step 3.3.1.1

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

$\frac{2 \cdot 3 x + 3 \cdot 9 - 2 \cdot 3 x - 2 \cdot - 7}{{(2 x + 9)}^{2}}$

Step 3.3.1.2

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

$\frac{3 \cdot 9 + 0 - 2 \cdot - 7}{{(2 x + 9)}^{2}}$

Step 3.3.1.3

Add $3 \cdot 9$ and $0$ .

$\frac{3 \cdot 9 - 2 \cdot - 7}{{(2 x + 9)}^{2}}$

Step 3.3.2

Simplify each term.

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Step 3.3.2.1

Multiply $3$ by $9$ .

$\frac{27 - 2 \cdot - 7}{{(2 x + 9)}^{2}}$

Step 3.3.2.2

Multiply $- 2$ by $- 7$ .

$\frac{27 + 14}{{(2 x + 9)}^{2}}$

Step 3.3.3

Add $27$ and $14$ .

$\frac{41}{{(2 x + 9)}^{2}}$