Calculus Examples

$y = \frac{8 x - 3}{6 x + 1}$

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 $f (x) = 8 x - 3$ and $g (x) = 6 x + 1$ .

$\frac{(6 x + 1) \frac{d}{d x} [8 x - 3] - (8 x - 3) \frac{d}{d x} [6 x + 1]}{{(6 x + 1)}^{2}}$

Step 2

Differentiate.

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Step 2.1

By the Sum Rule, the derivative of $8 x - 3$ with respect to $x$ is $\frac{d}{d x} [8 x] + \frac{d}{d x} [- 3]$ .

$\frac{(6 x + 1) (\frac{d}{d x} [8 x] + \frac{d}{d x} [- 3]) - (8 x - 3) \frac{d}{d x} [6 x + 1]}{{(6 x + 1)}^{2}}$

Step 2.2

Since $8$ is constant with respect to $x$ , the derivative of $8 x$ with respect to $x$ is $8 \frac{d}{d x} [x]$ .

$\frac{(6 x + 1) (8 \frac{d}{d x} [x] + \frac{d}{d x} [- 3]) - (8 x - 3) \frac{d}{d x} [6 x + 1]}{{(6 x + 1)}^{2}}$

Step 2.3

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

$\frac{(6 x + 1) (8 \cdot 1 + \frac{d}{d x} [- 3]) - (8 x - 3) \frac{d}{d x} [6 x + 1]}{{(6 x + 1)}^{2}}$

Step 2.4

Multiply $8$ by $1$ .

$\frac{(6 x + 1) (8 + \frac{d}{d x} [- 3]) - (8 x - 3) \frac{d}{d x} [6 x + 1]}{{(6 x + 1)}^{2}}$

Step 2.5

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

$\frac{(6 x + 1) (8 + 0) - (8 x - 3) \frac{d}{d x} [6 x + 1]}{{(6 x + 1)}^{2}}$

Step 2.6

Simplify the expression.

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Step 2.6.1

Add $8$ and $0$ .

$\frac{(6 x + 1) \cdot 8 - (8 x - 3) \frac{d}{d x} [6 x + 1]}{{(6 x + 1)}^{2}}$

Step 2.6.2

Move $8$ to the left of $6 x + 1$ .

$\frac{8 \cdot (6 x + 1) - (8 x - 3) \frac{d}{d x} [6 x + 1]}{{(6 x + 1)}^{2}}$

Step 2.7

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

$\frac{8 (6 x + 1) - (8 x - 3) (\frac{d}{d x} [6 x] + \frac{d}{d x} [1])}{{(6 x + 1)}^{2}}$

Step 2.8

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

$\frac{8 (6 x + 1) - (8 x - 3) (6 \frac{d}{d x} [x] + \frac{d}{d x} [1])}{{(6 x + 1)}^{2}}$

Step 2.9

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

$\frac{8 (6 x + 1) - (8 x - 3) (6 \cdot 1 + \frac{d}{d x} [1])}{{(6 x + 1)}^{2}}$

Step 2.10

Multiply $6$ by $1$ .

$\frac{8 (6 x + 1) - (8 x - 3) (6 + \frac{d}{d x} [1])}{{(6 x + 1)}^{2}}$

Step 2.11

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

$\frac{8 (6 x + 1) - (8 x - 3) (6 + 0)}{{(6 x + 1)}^{2}}$

Step 2.12

Simplify the expression.

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Step 2.12.1

Add $6$ and $0$ .

$\frac{8 (6 x + 1) - (8 x - 3) \cdot 6}{{(6 x + 1)}^{2}}$

Step 2.12.2

Multiply $6$ by $- 1$ .

$\frac{8 (6 x + 1) - 6 (8 x - 3)}{{(6 x + 1)}^{2}}$

Step 3

Simplify.

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Step 3.1

Apply the distributive property.

$\frac{8 (6 x) + 8 \cdot 1 - 6 (8 x - 3)}{{(6 x + 1)}^{2}}$

Step 3.2

Apply the distributive property.

$\frac{8 (6 x) + 8 \cdot 1 - 6 (8 x) - 6 \cdot - 3}{{(6 x + 1)}^{2}}$

Step 3.3

Simplify the numerator.

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Step 3.3.1

Combine the opposite terms in $8 (6 x) + 8 \cdot 1 - 6 (8 x) - 6 \cdot - 3$ .

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Step 3.3.1.1

Reorder the factors in the terms $8 (6 x)$ and $- 6 (8 x)$ .

$\frac{6 \cdot 8 x + 8 \cdot 1 - 6 \cdot 8 x - 6 \cdot - 3}{{(6 x + 1)}^{2}}$

Step 3.3.1.2

Subtract $6 \cdot 8 x$ from $6 \cdot 8 x$ .

$\frac{8 \cdot 1 + 0 - 6 \cdot - 3}{{(6 x + 1)}^{2}}$

Step 3.3.1.3

Add $8 \cdot 1$ and $0$ .

$\frac{8 \cdot 1 - 6 \cdot - 3}{{(6 x + 1)}^{2}}$

Step 3.3.2

Simplify each term.

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Step 3.3.2.1

Multiply $8$ by $1$ .

$\frac{8 - 6 \cdot - 3}{{(6 x + 1)}^{2}}$

Step 3.3.2.2

Multiply $- 6$ by $- 3$ .

$\frac{8 + 18}{{(6 x + 1)}^{2}}$

Step 3.3.3

Add $8$ and $18$ .

$\frac{26}{{(6 x + 1)}^{2}}$