Calculus Examples

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

Step 1

Find the derivative.

Tap for more steps...

Step 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 1.2

Differentiate.

Tap for more steps...

Step 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 1.2.2

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

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

Step 1.2.3

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 1.2.4

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 1.2.5

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 1.2.6

Simplify the expression.

Tap for more steps...

Step 1.2.6.1

Add $2 x$ and $0$ .

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

Step 1.2.6.2

Multiply $2$ by $- 1$ .

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

Step 1.3

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

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

Step 1.4

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

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

Step 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 1.6

Add $1$ and $1$ .

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

Step 1.7

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

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

Step 2

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

Tap for more steps...

Step 2.1

Set the numerator equal to zero.

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

Step 2.2

Solve the equation for $x$ .

Tap for more steps...

Step 2.2.1

Subtract $1$ from both sides of the equation.

$- x^{2} = - 1$

Step 2.2.2

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

Tap for more steps...

Step 2.2.2.1

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

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

Step 2.2.2.2

Simplify the left side.

Tap for more steps...

Step 2.2.2.2.1

Dividing two negative values results in a positive value.

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

Step 2.2.2.2.2

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

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

Step 2.2.2.3

Simplify the right side.

Tap for more steps...

Step 2.2.2.3.1

Divide $- 1$ by $- 1$ .

$x^{2} = 1$

Step 2.2.3

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

$x = \pm \sqrt{1}$

Step 2.2.4

Any root of $1$ is $1$ .

$x = \pm 1$

Step 2.2.5

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

Tap for more steps...

Step 2.2.5.1

First, use the positive value of the $\pm$ to find the first solution.

$x = 1$

Step 2.2.5.2

Next, use the negative value of the $\pm$ to find the second solution.

$x = - 1$

Step 2.2.5.3

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

$x = 1, - 1$

Step 3

Solve the original function $f (x) = \frac{x}{x^{2} + 1}$ at $x = 1$ .

Tap for more steps...

Step 3.1

Replace the variable $x$ with $1$ in the expression.

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

Step 3.2

Simplify the result.

Tap for more steps...

Step 3.2.1

Simplify the denominator.

Tap for more steps...

Step 3.2.1.1

One to any power is one.

$f (1) = \frac{1}{1 + 1}$

Step 3.2.1.2

Add $1$ and $1$ .

$f (1) = \frac{1}{2}$

Step 3.2.2

The final answer is $\frac{1}{2}$ .

$\frac{1}{2}$

Step 4

Solve the original function $f (x) = \frac{x}{x^{2} + 1}$ at $x = - 1$ .

Tap for more steps...

Step 4.1

Replace the variable $x$ with $- 1$ in the expression.

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

Step 4.2

Simplify the result.

Tap for more steps...

Step 4.2.1

Simplify the denominator.

Tap for more steps...

Step 4.2.1.1

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

$f (- 1) = \frac{- 1}{1 + 1}$

Step 4.2.1.2

Add $1$ and $1$ .

$f (- 1) = \frac{- 1}{2}$

Step 4.2.2

Move the negative in front of the fraction.

$f (- 1) = - \frac{1}{2}$

Step 4.2.3

The final answer is $- \frac{1}{2}$ .

$- \frac{1}{2}$

Step 5

The horizontal tangent lines on function $f (x) = \frac{x}{x^{2} + 1}$ are $y = \frac{1}{2}, y = - \frac{1}{2}$ .

$y = \frac{1}{2}, y = - \frac{1}{2}$

Step 6