Calculus Examples

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

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

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

Step 2

Differentiate.

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

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

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

Step 2.4

Multiply $2$ by $1$ .

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

Step 2.5

Since $- 3$ is constant with respect to $x$ , the derivative of $- 3$ with respect to $x$ is $0$ .

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

Step 2.6

Simplify the expression.

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Step 2.6.1

Add $2$ and $0$ .

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

Step 2.6.2

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

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

Step 2.7

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

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

Step 2.8

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

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

Step 2.9

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

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

Step 2.10

Multiply $3$ by $1$ .

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

Step 2.11

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

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

Step 2.12

Simplify the expression.

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Step 2.12.1

Add $3$ and $0$ .

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

Step 2.12.2

Multiply $3$ by $- 1$ .

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

Step 3

Simplify.

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Step 3.1

Apply the distributive property.

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

Step 3.2

Apply the distributive property.

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

Step 3.3

Simplify the numerator.

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Step 3.3.1

Simplify each term.

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Step 3.3.1.1

Multiply $3$ by $2$ .

$\frac{6 x + 2 \cdot 2 - 3 (2 x) - 3 \cdot - 3}{{(3 x + 2)}^{2}}$

Step 3.3.1.2

Multiply $2$ by $2$ .

$\frac{6 x + 4 - 3 (2 x) - 3 \cdot - 3}{{(3 x + 2)}^{2}}$

Step 3.3.1.3

Multiply $2$ by $- 3$ .

$\frac{6 x + 4 - 6 x - 3 \cdot - 3}{{(3 x + 2)}^{2}}$

Step 3.3.1.4

Multiply $- 3$ by $- 3$ .

$\frac{6 x + 4 - 6 x + 9}{{(3 x + 2)}^{2}}$

Step 3.3.2

Combine the opposite terms in $6 x + 4 - 6 x + 9$ .

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Step 3.3.2.1

Subtract $6 x$ from $6 x$ .

$\frac{0 + 4 + 9}{{(3 x + 2)}^{2}}$

Step 3.3.2.2

Add $0$ and $4$ .

$\frac{4 + 9}{{(3 x + 2)}^{2}}$

Step 3.3.3

Add $4$ and $9$ .

$\frac{13}{{(3 x + 2)}^{2}}$