Calculus Examples

$f (x) = \frac{x}{x^{2} + 1}$ on $- 3$ , $3$

Step 1

Find the critical points.

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Step 1.1

Find the first derivative.

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Step 1.1.1

Find the first derivative.

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Step 1.1.1.1

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

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

Step 1.1.1.2

Differentiate.

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Step 1.1.1.2.1

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

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

Step 1.1.1.2.2

Multiply $x^{2} + 1$ by $1$ .

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

Step 1.1.1.2.3

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

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

Step 1.1.1.2.4

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

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

Step 1.1.1.2.5

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

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

Step 1.1.1.2.6

Simplify the expression.

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Step 1.1.1.2.6.1

Add $2 x$ and $0$ .

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

Step 1.1.1.2.6.2

Multiply $2$ by $- 1$ .

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

Step 1.1.1.3

Raise $x$ to the power of $1$ .

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

Step 1.1.1.4

Raise $x$ to the power of $1$ .

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

Step 1.1.1.5

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 1.1.1.6

Add $1$ and $1$ .

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

Step 1.1.1.7

Subtract $2 x^{2}$ from $x^{2}$ .

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

Step 1.1.2

The first derivative of $f (x)$ with respect to $x$ is $\frac{- x^{2} + 1}{{(x^{2} + 1)}^{2}}$ .

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

Step 1.2

Set the first derivative equal to $0$ then solve the equation $\frac{- x^{2} + 1}{{(x^{2} + 1)}^{2}} = 0$ .

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Step 1.2.1

Set the first derivative equal to $0$ .

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

Step 1.2.2

Set the numerator equal to zero.

$- x^{2} + 1 = 0$

Step 1.2.3

Solve the equation for $x$ .

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Step 1.2.3.1

Subtract $1$ from both sides of the equation.

$- x^{2} = - 1$

Step 1.2.3.2

Divide each term in $- x^{2} = - 1$ by $- 1$ and simplify.

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Step 1.2.3.2.1

Divide each term in $- x^{2} = - 1$ by $- 1$ .

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

Step 1.2.3.2.2

Simplify the left side.

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Step 1.2.3.2.2.1

Dividing two negative values results in a positive value.

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

Step 1.2.3.2.2.2

Divide $x^{2}$ by $1$ .

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

Step 1.2.3.2.3

Simplify the right side.

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Step 1.2.3.2.3.1

Divide $- 1$ by $- 1$ .

$x^{2} = 1$

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.

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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.

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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 $\frac{x}{x^{2} + 1}$ at each $x$ value where the derivative is $0$ or undefined.

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Step 1.4.1

Evaluate at $x = 1$ .

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Step 1.4.1.1

Substitute $1$ for $x$ .

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

Step 1.4.1.2

Simplify the denominator.

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Step 1.4.1.2.1

One to any power is one.

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

Step 1.4.1.2.2

Add $1$ and $1$ .

$\frac{1}{2}$

Step 1.4.2

Evaluate at $x = - 1$ .

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Step 1.4.2.1

Substitute $- 1$ for $x$ .

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

Step 1.4.2.2

Simplify.

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Step 1.4.2.2.1

Simplify the denominator.

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Step 1.4.2.2.1.1

Raise $- 1$ to the power of $2$ .

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

Step 1.4.2.2.1.2

Add $1$ and $1$ .

$\frac{- 1}{2}$

Step 1.4.2.2.2

Move the negative in front of the fraction.

$- \frac{1}{2}$

Step 1.4.3

List all of the points.

$(1, \frac{1}{2}), (- 1, - \frac{1}{2})$

Step 2

Evaluate at the included endpoints.

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Step 2.1

Evaluate at $x = - 3$ .

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Step 2.1.1

Substitute $- 3$ for $x$ .

$\frac{- 3}{{(- 3)}^{2} + 1}$

Step 2.1.2

Simplify.

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Step 2.1.2.1

Simplify the denominator.

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Step 2.1.2.1.1

Raise $- 3$ to the power of $2$ .

$\frac{- 3}{9 + 1}$

Step 2.1.2.1.2

Add $9$ and $1$ .

$\frac{- 3}{10}$

Step 2.1.2.2

Move the negative in front of the fraction.

$- \frac{3}{10}$

Step 2.2

Evaluate at $x = 3$ .

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Step 2.2.1

Substitute $3$ for $x$ .

$\frac{3}{{(3)}^{2} + 1}$

Step 2.2.2

Simplify the denominator.

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Step 2.2.2.1

Raise $3$ to the power of $2$ .

$\frac{3}{9 + 1}$

Step 2.2.2.2

Add $9$ and $1$ .

$\frac{3}{10}$

Step 2.3

List all of the points.

$(- 3, - \frac{3}{10}), (3, \frac{3}{10})$

Step 3

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: $(1, \frac{1}{2})$

Absolute Minimum: $(- 1, - \frac{1}{2})$

Step 4