Calculus Examples

\frac{x}{x - 1}

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

Step 1

Differentiate using the Quotient Rule which states that

\frac{d}{d x} [\frac{f (x)}{g (x)}]

$\frac{d}{d x} [\frac{f (x)}{g (x)}]$ is

\frac{g (x) \frac{d}{d x} [f (x)] - f (x) \frac{d}{d x} [g (x)]}{{g (x)}^{2}}

$\frac{g (x) \frac{d}{d x} [f (x)] - f (x) \frac{d}{d x} [g (x)]}{{g (x)}^{2}}$ where

f (x) = x

$f (x) = x$ and

g (x) = x - 1

$g (x) = x - 1$ .

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

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

Step 2

Differentiate.

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Step 2.1

Differentiate using the Power Rule which states that

\frac{d}{d x} [x^{n}]

$\frac{d}{d x} [x^{n}]$ is

n x^{n - 1}

$n x^{n - 1}$ where

n = 1

$n = 1$ .

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

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

Step 2.2

Multiply

x - 1

$x - 1$ by

1

$1$ .

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

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

Step 2.3

By the Sum Rule, the derivative of

x - 1

$x - 1$ with respect to

x

$x$ is

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

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

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

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

Step 2.4

Differentiate using the Power Rule which states that

\frac{d}{d x} [x^{n}]

$\frac{d}{d x} [x^{n}]$ is

n x^{n - 1}

$n x^{n - 1}$ where

n = 1

$n = 1$ .

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

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

Step 2.5

Since

- 1

$- 1$ is constant with respect to

x

$x$ , the derivative of

- 1

$- 1$ with respect to

x

$x$ is

0

$0$ .

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

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

Step 2.6

Simplify by adding terms.

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Step 2.6.1

Add

1

$1$ and

0

$0$ .

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

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

Step 2.6.2

Multiply

- 1

$- 1$ by

1

$1$ .

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

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

Step 2.6.3

Subtract

x

$x$ from

x

$x$ .

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

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

Step 2.6.4

Simplify the expression.

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Step 2.6.4.1

Subtract

1

$1$ from

0

$0$ .

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

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

Step 2.6.4.2

Move the negative in front of the fraction.

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

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

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

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

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

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

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

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