Calculus Examples

$\frac{4 x - 1}{4 x + 1}$

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

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

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 $\frac{d}{d x} [4 x] + \frac{d}{d x} [- 1]$ .

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

Step 2.2

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

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

Step 2.3

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

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

Step 2.4

Multiply $4$ by $1$ .

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

Step 2.5

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

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

Step 2.6

Simplify the expression.

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Step 2.6.1

Add $4$ and $0$ .

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

Step 2.6.2

Move $4$ to the left of $4 x + 1$ .

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

Step 2.7

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

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

Step 2.8

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

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

Step 2.9

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

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

Step 2.10

Multiply $4$ by $1$ .

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

Step 2.11

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

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

Step 2.12

Simplify the expression.

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Step 2.12.1

Add $4$ and $0$ .

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

Step 2.12.2

Multiply $4$ by $- 1$ .

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

Step 3

Simplify.

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Step 3.1

Apply the distributive property.

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

Step 3.2

Apply the distributive property.

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

Step 3.3

Simplify the numerator.

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Step 3.3.1

Combine the opposite terms in $4 (4 x) + 4 \cdot 1 - 4 (4 x) - 4 \cdot - 1$ .

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Step 3.3.1.1

Subtract $4 (4 x)$ from $4 (4 x)$ .

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

Step 3.3.1.2

Add $4 \cdot 1$ and $0$ .

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

Step 3.3.2

Simplify each term.

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Step 3.3.2.1

Multiply $4$ by $1$ .

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

Step 3.3.2.2

Multiply $- 4$ by $- 1$ .

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

Step 3.3.3

Add $4$ and $4$ .

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