Calculus Examples

$y = \frac{4 x - 1}{8 x + 7}$

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 $f (x) = 4 x - 1$ and $g (x) = 8 x + 7$ .

$\frac{(8 x + 7) \frac{d}{d x} [4 x - 1] - (4 x - 1) \frac{d}{d x} [8 x + 7]}{{(8 x + 7)}^{2}}$

Step 2

Differentiate.

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Step 2.1

By the Sum Rule, the derivative of $4 x - 1$ with respect to $x$ is $\frac{d}{d x} [4 x] + \frac{d}{d x} [- 1]$ .

$\frac{(8 x + 7) (\frac{d}{d x} [4 x] + \frac{d}{d x} [- 1]) - (4 x - 1) \frac{d}{d x} [8 x + 7]}{{(8 x + 7)}^{2}}$

Step 2.2

Since $4$ is constant with respect to $x$ , the derivative of $4 x$ with respect to $x$ is $4 \frac{d}{d x} [x]$ .

$\frac{(8 x + 7) (4 \frac{d}{d x} [x] + \frac{d}{d x} [- 1]) - (4 x - 1) \frac{d}{d x} [8 x + 7]}{{(8 x + 7)}^{2}}$

Step 2.3

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

$\frac{(8 x + 7) (4 \cdot 1 + \frac{d}{d x} [- 1]) - (4 x - 1) \frac{d}{d x} [8 x + 7]}{{(8 x + 7)}^{2}}$

Step 2.4

Multiply $4$ by $1$ .

$\frac{(8 x + 7) (4 + \frac{d}{d x} [- 1]) - (4 x - 1) \frac{d}{d x} [8 x + 7]}{{(8 x + 7)}^{2}}$

Step 2.5

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

$\frac{(8 x + 7) (4 + 0) - (4 x - 1) \frac{d}{d x} [8 x + 7]}{{(8 x + 7)}^{2}}$

Step 2.6

Simplify the expression.

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Step 2.6.1

Add $4$ and $0$ .

$\frac{(8 x + 7) \cdot 4 - (4 x - 1) \frac{d}{d x} [8 x + 7]}{{(8 x + 7)}^{2}}$

Step 2.6.2

Move $4$ to the left of $8 x + 7$ .

$\frac{4 \cdot (8 x + 7) - (4 x - 1) \frac{d}{d x} [8 x + 7]}{{(8 x + 7)}^{2}}$

Step 2.7

By the Sum Rule, the derivative of $8 x + 7$ with respect to $x$ is $\frac{d}{d x} [8 x] + \frac{d}{d x} [7]$ .

$\frac{4 (8 x + 7) - (4 x - 1) (\frac{d}{d x} [8 x] + \frac{d}{d x} [7])}{{(8 x + 7)}^{2}}$

Step 2.8

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

$\frac{4 (8 x + 7) - (4 x - 1) (8 \frac{d}{d x} [x] + \frac{d}{d x} [7])}{{(8 x + 7)}^{2}}$

Step 2.9

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

$\frac{4 (8 x + 7) - (4 x - 1) (8 \cdot 1 + \frac{d}{d x} [7])}{{(8 x + 7)}^{2}}$

Step 2.10

Multiply $8$ by $1$ .

$\frac{4 (8 x + 7) - (4 x - 1) (8 + \frac{d}{d x} [7])}{{(8 x + 7)}^{2}}$

Step 2.11

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

$\frac{4 (8 x + 7) - (4 x - 1) (8 + 0)}{{(8 x + 7)}^{2}}$

Step 2.12

Simplify the expression.

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Step 2.12.1

Add $8$ and $0$ .

$\frac{4 (8 x + 7) - (4 x - 1) \cdot 8}{{(8 x + 7)}^{2}}$

Step 2.12.2

Multiply $8$ by $- 1$ .

$\frac{4 (8 x + 7) - 8 (4 x - 1)}{{(8 x + 7)}^{2}}$

Step 3

Simplify.

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Step 3.1

Apply the distributive property.

$\frac{4 (8 x) + 4 \cdot 7 - 8 (4 x - 1)}{{(8 x + 7)}^{2}}$

Step 3.2

Apply the distributive property.

$\frac{4 (8 x) + 4 \cdot 7 - 8 (4 x) - 8 \cdot - 1}{{(8 x + 7)}^{2}}$

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 $8$ by $4$ .

$\frac{32 x + 4 \cdot 7 - 8 (4 x) - 8 \cdot - 1}{{(8 x + 7)}^{2}}$

Step 3.3.1.2

Multiply $4$ by $7$ .

$\frac{32 x + 28 - 8 (4 x) - 8 \cdot - 1}{{(8 x + 7)}^{2}}$

Step 3.3.1.3

Multiply $4$ by $- 8$ .

$\frac{32 x + 28 - 32 x - 8 \cdot - 1}{{(8 x + 7)}^{2}}$

Step 3.3.1.4

Multiply $- 8$ by $- 1$ .

$\frac{32 x + 28 - 32 x + 8}{{(8 x + 7)}^{2}}$

Step 3.3.2

Combine the opposite terms in $32 x + 28 - 32 x + 8$ .

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Step 3.3.2.1

Subtract $32 x$ from $32 x$ .

$\frac{0 + 28 + 8}{{(8 x + 7)}^{2}}$

Step 3.3.2.2

Add $0$ and $28$ .

$\frac{28 + 8}{{(8 x + 7)}^{2}}$

Step 3.3.3

Add $28$ and $8$ .

$\frac{36}{{(8 x + 7)}^{2}}$