Calculus Examples

\frac{x}{x^{2} + 1}

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

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^{2} + 1

$g (x) = x^{2} + 1$ .

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

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

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

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

Step 2.2

Multiply

x^{2} + 1

$x^{2} + 1$ by

1

$1$ .

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

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

Step 2.3

By the Sum Rule, the derivative of

x^{2} + 1

$x^{2} + 1$ with respect to

x

$x$ is

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

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

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

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

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 = 2

$n = 2$ .

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

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

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

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

Step 2.6

Simplify the expression.

Tap for more steps...

Step 2.6.1

Add

2 x

$2 x$ and

0

$0$ .

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

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

Step 2.6.2

Multiply

2

$2$ by

- 1

$- 1$ .

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

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

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

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

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

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

Step 3

Raise

x

$x$ to the power of

1

$1$ .

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

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

Step 4

Raise

x

$x$ to the power of

1

$1$ .

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

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

Step 5

Use the power rule

a^{m} a^{n} = a^{m + n}

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

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

Step 6

Add

1

$1$ and

1

$1$ .

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

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

Step 7

Subtract

2 x^{2}

$2 x^{2}$ from

x^{2}

$x^{2}$ .

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

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