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.

Tap for more steps...

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.

Tap for more steps...

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

Add

0

$0$ and

1

$1$ .

\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}}

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