Calculus Examples

$\frac{6 x + 1}{3 x - 7}$

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 $f (x) = 6 x + 1$ and $g (x) = 3 x - 7$ .

$\frac{(3 x - 7) \frac{d}{d x} [6 x + 1] - (6 x + 1) \frac{d}{d x} [3 x - 7]}{{(3 x - 7)}^{2}}$

Step 2

Differentiate.

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Step 2.1

By the Sum Rule, the derivative of $6 x + 1$ with respect to $x$ is $\frac{d}{d x} [6 x] + \frac{d}{d x} [1]$ .

$\frac{(3 x - 7) (\frac{d}{d x} [6 x] + \frac{d}{d x} [1]) - (6 x + 1) \frac{d}{d x} [3 x - 7]}{{(3 x - 7)}^{2}}$

Step 2.2

Since $6$ is constant with respect to $x$ , the derivative of $6 x$ with respect to $x$ is $6 \frac{d}{d x} [x]$ .

$\frac{(3 x - 7) (6 \frac{d}{d x} [x] + \frac{d}{d x} [1]) - (6 x + 1) \frac{d}{d x} [3 x - 7]}{{(3 x - 7)}^{2}}$

Step 2.3

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

$\frac{(3 x - 7) (6 \cdot 1 + \frac{d}{d x} [1]) - (6 x + 1) \frac{d}{d x} [3 x - 7]}{{(3 x - 7)}^{2}}$

Step 2.4

Multiply $6$ by $1$ .

$\frac{(3 x - 7) (6 + \frac{d}{d x} [1]) - (6 x + 1) \frac{d}{d x} [3 x - 7]}{{(3 x - 7)}^{2}}$

Step 2.5

Since $1$ is constant with respect to $x$ , the derivative of $1$ with respect to $x$ is $0$ .

$\frac{(3 x - 7) (6 + 0) - (6 x + 1) \frac{d}{d x} [3 x - 7]}{{(3 x - 7)}^{2}}$

Step 2.6

Simplify the expression.

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Step 2.6.1

Add $6$ and $0$ .

$\frac{(3 x - 7) \cdot 6 - (6 x + 1) \frac{d}{d x} [3 x - 7]}{{(3 x - 7)}^{2}}$

Step 2.6.2

Move $6$ to the left of $3 x - 7$ .

$\frac{6 \cdot (3 x - 7) - (6 x + 1) \frac{d}{d x} [3 x - 7]}{{(3 x - 7)}^{2}}$

Step 2.7

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

$\frac{6 (3 x - 7) - (6 x + 1) (\frac{d}{d x} [3 x] + \frac{d}{d x} [- 7])}{{(3 x - 7)}^{2}}$

Step 2.8

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

$\frac{6 (3 x - 7) - (6 x + 1) (3 \frac{d}{d x} [x] + \frac{d}{d x} [- 7])}{{(3 x - 7)}^{2}}$

Step 2.9

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

$\frac{6 (3 x - 7) - (6 x + 1) (3 \cdot 1 + \frac{d}{d x} [- 7])}{{(3 x - 7)}^{2}}$

Step 2.10

Multiply $3$ by $1$ .

$\frac{6 (3 x - 7) - (6 x + 1) (3 + \frac{d}{d x} [- 7])}{{(3 x - 7)}^{2}}$

Step 2.11

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

$\frac{6 (3 x - 7) - (6 x + 1) (3 + 0)}{{(3 x - 7)}^{2}}$

Step 2.12

Simplify the expression.

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Step 2.12.1

Add $3$ and $0$ .

$\frac{6 (3 x - 7) - (6 x + 1) \cdot 3}{{(3 x - 7)}^{2}}$

Step 2.12.2

Multiply $3$ by $- 1$ .

$\frac{6 (3 x - 7) - 3 (6 x + 1)}{{(3 x - 7)}^{2}}$

Step 3

Simplify.

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Step 3.1

Apply the distributive property.

$\frac{6 (3 x) + 6 \cdot - 7 - 3 (6 x + 1)}{{(3 x - 7)}^{2}}$

Step 3.2

Apply the distributive property.

$\frac{6 (3 x) + 6 \cdot - 7 - 3 (6 x) - 3 \cdot 1}{{(3 x - 7)}^{2}}$

Step 3.3

Simplify the numerator.

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Step 3.3.1

Combine the opposite terms in $6 (3 x) + 6 \cdot - 7 - 3 (6 x) - 3 \cdot 1$ .

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Step 3.3.1.1

Reorder the factors in the terms $6 (3 x)$ and $- 3 (6 x)$ .

$\frac{3 \cdot 6 x + 6 \cdot - 7 - 3 \cdot 6 x - 3 \cdot 1}{{(3 x - 7)}^{2}}$

Step 3.3.1.2

Subtract $3 \cdot 6 x$ from $3 \cdot 6 x$ .

$\frac{6 \cdot - 7 + 0 - 3 \cdot 1}{{(3 x - 7)}^{2}}$

Step 3.3.1.3

Add $6 \cdot - 7$ and $0$ .

$\frac{6 \cdot - 7 - 3 \cdot 1}{{(3 x - 7)}^{2}}$

Step 3.3.2

Simplify each term.

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Step 3.3.2.1

Multiply $6$ by $- 7$ .

$\frac{- 42 - 3 \cdot 1}{{(3 x - 7)}^{2}}$

Step 3.3.2.2

Multiply $- 3$ by $1$ .

$\frac{- 42 - 3}{{(3 x - 7)}^{2}}$

Step 3.3.3

Subtract $3$ from $- 42$ .

$\frac{- 45}{{(3 x - 7)}^{2}}$

Step 3.4

Move the negative in front of the fraction.

$- \frac{45}{{(3 x - 7)}^{2}}$