Calculus Examples

f (x) = \frac{4 x}{x^{2} + 1}

$f (x) = \frac{4 x}{x^{2} + 1}$ on

- 4

$- 4$ ,

0

$0$

Step 1

Find the critical points.

Tap for more steps...

Step 1.1

Find the first derivative.

Tap for more steps...

Step 1.1.1

Find the first derivative.

Tap for more steps...

Step 1.1.1.1

Since

4

$4$ is constant with respect to

x

$x$ , the derivative of

\frac{4 x}{x^{2} + 1}

$\frac{4 x}{x^{2} + 1}$ with respect to

x

$x$ is

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

$4 \frac{d}{d x} [\frac{x}{x^{2} + 1}]$ .

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

$4 \frac{d}{d x} [\frac{x}{x^{2} + 1}]$

Step 1.1.1.2

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

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

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

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

Step 1.1.1.3

Differentiate.

Tap for more steps...

Step 1.1.1.3.1

Differentiate using the Power Rule which states that

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

$n x^{n - 1}$ where

$n = 1$ .

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

Step 1.1.1.3.2

Multiply

$x^{2} + 1$ by

$1$ .

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

Step 1.1.1.3.3

By the Sum Rule, the derivative of

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

$x$ is

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

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

Step 1.1.1.3.4

Differentiate using the Power Rule which states that

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

$n x^{n - 1}$ where

$n = 2$ .

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

Step 1.1.1.3.5

Since

$1$ is constant with respect to

$x$ , the derivative of

$1$ with respect to

$x$ is

$0$ .

$4 \frac{x^{2} + 1 - x (2 x + 0)}{{(x^{2} + 1)}^{2}}$

Step 1.1.1.3.6

Simplify the expression.

Tap for more steps...

Step 1.1.1.3.6.1

Add

$2 x$ and

$0$ .

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

Step 1.1.1.3.6.2

Multiply

$2$ by

$- 1$ .

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

Step 1.1.1.4

Raise

$x$ to the power of

$1$ .

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

Step 1.1.1.5

Raise

$x$ to the power of

$1$ .

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

Step 1.1.1.6

Use the power rule

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

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

Step 1.1.1.7

Add

$1$ and

$1$ .

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

Step 1.1.1.8

Subtract

$2 x^{2}$ from

$x^{2}$ .

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

Step 1.1.1.9

Combine

$4$ and

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

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

Step 1.1.1.10

Simplify.

Tap for more steps...

Step 1.1.1.10.1

Apply the distributive property.

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

Step 1.1.1.10.2

Simplify each term.

Tap for more steps...

Step 1.1.1.10.2.1

Multiply

$- 1$ by

$4$ .

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

Step 1.1.1.10.2.2

Multiply

$4$ by

$1$ .

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

Step 1.1.2

The first derivative of

$f (x)$ with respect to

$x$ is

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

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

Step 1.2

Set the first derivative equal to

$0$ then solve the equation

$\frac{- 4 x^{2} + 4}{{(x^{2} + 1)}^{2}} = 0$ .

Tap for more steps...

Step 1.2.1

Set the first derivative equal to

$0$ .

$\frac{- 4 x^{2} + 4}{{(x^{2} + 1)}^{2}} = 0$

Step 1.2.2

Set the numerator equal to zero.

$- 4 x^{2} + 4 = 0$

Step 1.2.3

Solve the equation for

$x$ .

Tap for more steps...

Step 1.2.3.1

Subtract

$4$ from both sides of the equation.

$- 4 x^{2} = - 4$

Step 1.2.3.2

Divide each term in

$- 4 x^{2} = - 4$ by

$- 4$ and simplify.

Tap for more steps...

Step 1.2.3.2.1

Divide each term in

$- 4 x^{2} = - 4$ by

$- 4$ .

$\frac{- 4 x^{2}}{- 4} = \frac{- 4}{- 4}$

Step 1.2.3.2.2

Simplify the left side.

Tap for more steps...

Step 1.2.3.2.2.1

Cancel the common factor of

$- 4$ .

Tap for more steps...

Step 1.2.3.2.2.1.1

Cancel the common factor.

$\frac{- 4 x^{2}}{- 4} = \frac{- 4}{- 4}$

Step 1.2.3.2.2.1.2

Divide

$x^{2}$ by

$1$ .

$x^{2} = \frac{- 4}{- 4}$

Step 1.2.3.2.3

Simplify the right side.

Tap for more steps...

Step 1.2.3.2.3.1

Divide

$- 4$ by

$- 4$ .

$x^{2} = 1$

Step 1.2.3.3

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x = \pm \sqrt{1}$

Step 1.2.3.4

Any root of

$1$ is

$1$ .

$x = \pm 1$

Step 1.2.3.5

The complete solution is the result of both the positive and negative portions of the solution.

Tap for more steps...

Step 1.2.3.5.1

First, use the positive value of the

$\pm$ to find the first solution.

$x = 1$

Step 1.2.3.5.2

Next, use the negative value of the

$\pm$ to find the second solution.

$x = - 1$

Step 1.2.3.5.3

The complete solution is the result of both the positive and negative portions of the solution.

$x = 1, - 1$

Step 1.3

Find the values where the derivative is undefined.

Tap for more steps...

Step 1.3.1

The domain of the expression is all real numbers except where the expression is undefined. In this case, there is no real number that makes the expression undefined.

Step 1.4

Evaluate

$\frac{4 x}{x^{2} + 1}$ at each

$x$ value where the derivative is

$0$ or undefined.

Tap for more steps...

Step 1.4.1

Evaluate at

$x = 1$ .

Tap for more steps...

Step 1.4.1.1

Substitute

$1$ for

$x$ .

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

Step 1.4.1.2

Simplify.

Tap for more steps...

Step 1.4.1.2.1

Multiply

$4$ by

$1$ .

$\frac{4}{1^{2} + 1}$

Step 1.4.1.2.2

Simplify the denominator.

Tap for more steps...

Step 1.4.1.2.2.1

One to any power is one.

$\frac{4}{1 + 1}$

Step 1.4.1.2.2.2

Add

$1$ and

$1$ .

$\frac{4}{2}$

Step 1.4.1.2.3

Divide

$4$ by

$2$ .

$2$

Step 1.4.2

Evaluate at

$x = - 1$ .

Tap for more steps...

Step 1.4.2.1

Substitute

$- 1$ for

$x$ .

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

Step 1.4.2.2

Simplify.

Tap for more steps...

Step 1.4.2.2.1

Multiply

$4$ by

$- 1$ .

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

Step 1.4.2.2.2

Simplify the denominator.

Tap for more steps...

Step 1.4.2.2.2.1

Raise

$- 1$ to the power of

$2$ .

$\frac{- 4}{1 + 1}$

Step 1.4.2.2.2.2

Add

$1$ and

$1$ .

$\frac{- 4}{2}$

Step 1.4.2.2.3

Divide

$- 4$ by

$2$ .

$- 2$

Step 1.4.3

List all of the points.

$(1, 2), (- 1, - 2)$

Step 2

Exclude the points that are not on the interval.

$(- 1, - 2)$

Step 3

Evaluate at the included endpoints.

Tap for more steps...

Step 3.1

Evaluate at

$x = - 4$ .

Tap for more steps...

Step 3.1.1

Substitute

$- 4$ for

$x$ .

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

Step 3.1.2

Simplify.

Tap for more steps...

Step 3.1.2.1

Multiply

$4$ by

$- 4$ .

$\frac{- 16}{{(- 4)}^{2} + 1}$

Step 3.1.2.2

Simplify the denominator.

Tap for more steps...

Step 3.1.2.2.1

Raise

$- 4$ to the power of

$2$ .

$\frac{- 16}{16 + 1}$

Step 3.1.2.2.2

Add

$16$ and

$1$ .

$\frac{- 16}{17}$

Step 3.1.2.3

Move the negative in front of the fraction.

$- \frac{16}{17}$

Step 3.2

Evaluate at

$x = 0$ .

Tap for more steps...

Step 3.2.1

Substitute

$0$ for

$x$ .

$\frac{4 (0)}{{(0)}^{2} + 1}$

Step 3.2.2

Simplify.

Tap for more steps...

Step 3.2.2.1

Multiply

$4$ by

$0$ .

$\frac{0}{0^{2} + 1}$

Step 3.2.2.2

Simplify the denominator.

Tap for more steps...

Step 3.2.2.2.1

Raising

$0$ to any positive power yields

$0$ .

$\frac{0}{0 + 1}$

Step 3.2.2.2.2

Add

$0$ and

$1$ .

$\frac{0}{1}$

Step 3.2.2.3

Divide

$0$ by

$1$ .

$0$

Step 3.3

List all of the points.

$(- 4, - \frac{16}{17}), (0, 0)$

Step 4

Compare the

$f (x)$ values found for each value of

$x$ in order to determine the absolute maximum and minimum over the given interval. The maximum will occur at the highest

$f (x)$ value and the minimum will occur at the lowest

$f (x)$ value.

Absolute Maximum:

$(0, 0)$

Absolute Minimum:

$(- 1, - 2)$

Step 5