Calculus Examples

$\frac{4 x + 2}{4 x - 2}$

Step 1

Cancel the common factor of $4 x + 2$ and $4 x - 2$ .

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Step 1.1

Factor $2$ out of $4 x$ .

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

Step 1.2

Factor $2$ out of $2$ .

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

Step 1.3

Factor $2$ out of $2 (2 x) + 2 (1)$ .

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

Step 1.4

Cancel the common factors.

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Step 1.4.1

Factor $2$ out of $4 x$ .

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

Step 1.4.2

Factor $2$ out of $- 2$ .

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

Step 1.4.3

Factor $2$ out of $2 (2 x) + 2 (- 1)$ .

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

Step 1.4.4

Cancel the common factor.

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

Step 1.4.5

Rewrite the expression.

$\frac{d}{d x} [\frac{2 x + 1}{2 x - 1}]$

Step 2

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 $f (x) = 2 x + 1$ and $g (x) = 2 x - 1$ .

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

Step 3

Differentiate.

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Step 3.1

By the Sum Rule, the derivative of $2 x + 1$ with respect to $x$ is $\frac{d}{d x} [2 x] + \frac{d}{d x} [1]$ .

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

Step 3.2

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

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

Step 3.3

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

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

Step 3.4

Multiply $2$ by $1$ .

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

Step 3.5

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

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

Step 3.6

Simplify the expression.

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Step 3.6.1

Add $2$ and $0$ .

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

Step 3.6.2

Move $2$ to the left of $2 x - 1$ .

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

Step 3.7

By the Sum Rule, the derivative of $2 x - 1$ with respect to $x$ is $\frac{d}{d x} [2 x] + \frac{d}{d x} [- 1]$ .

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

Step 3.8

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

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

Step 3.9

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

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

Step 3.10

Multiply $2$ by $1$ .

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

Step 3.11

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

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

Step 3.12

Simplify the expression.

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Step 3.12.1

Add $2$ and $0$ .

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

Step 3.12.2

Multiply $2$ by $- 1$ .

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

Step 4

Simplify.

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Step 4.1

Apply the distributive property.

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

Step 4.2

Apply the distributive property.

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

Step 4.3

Simplify the numerator.

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Step 4.3.1

Combine the opposite terms in $2 (2 x) + 2 \cdot - 1 - 2 (2 x) - 2 \cdot 1$ .

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Step 4.3.1.1

Subtract $2 (2 x)$ from $2 (2 x)$ .

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

Step 4.3.1.2

Add $2 \cdot - 1$ and $0$ .

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

Step 4.3.2

Simplify each term.

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Step 4.3.2.1

Multiply $2$ by $- 1$ .

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

Step 4.3.2.2

Multiply $- 2$ by $1$ .

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

Step 4.3.3

Subtract $2$ from $- 2$ .

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

Step 4.4

Move the negative in front of the fraction.

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