Calculus Examples

$y = \frac{x}{2 x - 1}$

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

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

Step 2

Differentiate.

Tap for more steps...

Step 2.1

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

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

Step 2.2

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

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

Step 2.3

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

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

Step 2.4

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

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

Step 2.5

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

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

Step 2.6

Multiply $2$ by $1$ .

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

Step 2.7

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

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

Step 2.8

Simplify by adding terms.

Tap for more steps...

Step 2.8.1

Add $2$ and $0$ .

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

Step 2.8.2

Multiply $2$ by $- 1$ .

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

Step 2.8.3

Subtract $2 x$ from $2 x$ .

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

Step 2.8.4

Simplify the expression.

Tap for more steps...

Step 2.8.4.1

Subtract $1$ from $0$ .

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

Step 2.8.4.2

Move the negative in front of the fraction.

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