Calculus Examples

$P (x) = (\frac{4 x - 8}{8 x - 4})$

Step 1

Reduce the expression by cancelling the common factors.

Tap for more steps...

Step 1.1

Cancel the common factor of $4 x - 8$ and $8 x - 4$ .

Tap for more steps...

Step 1.1.1

Factor $4$ out of $4 x$ .

$\frac{d}{d x} [(\frac{4 (x) - 8}{8 x - 4})]$

Step 1.1.2

Factor $4$ out of $- 8$ .

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

Step 1.1.3

Factor $4$ out of $4 (x) + 4 (- 2)$ .

$\frac{d}{d x} [(\frac{4 (x - 2)}{8 x - 4})]$

Step 1.1.4

Cancel the common factors.

Tap for more steps...

Step 1.1.4.1

Factor $4$ out of $8 x$ .

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

Step 1.1.4.2

Factor $4$ out of $- 4$ .

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

Step 1.1.4.3

Factor $4$ out of $4 (2 x) + 4 (- 1)$ .

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

Step 1.1.4.4

Cancel the common factor.

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

Step 1.1.4.5

Rewrite the expression.

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

Step 1.2

Remove parentheses.

Tap for more steps...

Step 1.2.1

Factor $4$ out of $4 x$ .

$\frac{d}{d x} [\frac{4 (x) - 8}{8 x - 4}]$

Step 1.2.2

Factor $4$ out of $- 8$ .

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

Step 1.2.3

Factor $4$ out of $4 (x) + 4 (- 2)$ .

$\frac{d}{d x} [\frac{4 (x - 2)}{8 x - 4}]$

Step 1.2.4

Cancel the common factors.

Tap for more steps...

Step 1.2.4.1

Factor $4$ out of $8 x$ .

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

Step 1.2.4.2

Factor $4$ out of $- 4$ .

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

Step 1.2.4.3

Factor $4$ out of $4 (2 x) + 4 (- 1)$ .

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

Step 1.2.4.4

Cancel the common factor.

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

Step 1.2.4.5

Rewrite the expression.

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

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

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

Step 3

Differentiate.

Tap for more steps...

Step 3.1

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

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

Step 3.2

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

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

Step 3.3

Since $- 2$ is constant with respect to $x$ , the derivative of $- 2$ with respect to $x$ is $0$ .

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

Step 3.4

Simplify the expression.

Tap for more steps...

Step 3.4.1

Add $1$ and $0$ .

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

Step 3.4.2

Multiply $2 x - 1$ by $1$ .

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

Step 3.5

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

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

Step 3.6

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

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

Step 3.7

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

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

Step 3.8

Multiply $2$ by $1$ .

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

Step 3.9

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

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

Step 3.10

Simplify the expression.

Tap for more steps...

Step 3.10.1

Add $2$ and $0$ .

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

Step 3.10.2

Multiply $2$ by $- 1$ .

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

Step 4

Simplify.

Tap for more steps...

Step 4.1

Apply the distributive property.

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

Step 4.2

Simplify the numerator.

Tap for more steps...

Step 4.2.1

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

Tap for more steps...

Step 4.2.1.1

Subtract $2 x$ from $2 x$ .

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

Step 4.2.1.2

Subtract $1$ from $0$ .

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

Step 4.2.2

Multiply $- 2$ by $- 2$ .

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

Step 4.2.3

Add $- 1$ and $4$ .

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