Calculus Examples

$f (x) = \frac{x}{x^{2} + 9}$

Step 1

Find the derivative.

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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 $g (x) = x^{2} + 9$ .

$\frac{(x^{2} + 9) \frac{d}{d x} [x] - x \frac{d}{d x} [x^{2} + 9]}{{(x^{2} + 9)}^{2}}$

Step 1.2

Differentiate.

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Step 1.2.1

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

$\frac{(x^{2} + 9) \cdot 1 - x \frac{d}{d x} [x^{2} + 9]}{{(x^{2} + 9)}^{2}}$

Step 1.2.2

Multiply $x^{2} + 9$ by $1$ .

$\frac{x^{2} + 9 - x \frac{d}{d x} [x^{2} + 9]}{{(x^{2} + 9)}^{2}}$

Step 1.2.3

By the Sum Rule, the derivative of $x^{2} + 9$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [9]$ .

$\frac{x^{2} + 9 - x (\frac{d}{d x} [x^{2}] + \frac{d}{d x} [9])}{{(x^{2} + 9)}^{2}}$

Step 1.2.4

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

$\frac{x^{2} + 9 - x (2 x + \frac{d}{d x} [9])}{{(x^{2} + 9)}^{2}}$

Step 1.2.5

Since $9$ is constant with respect to $x$ , the derivative of $9$ with respect to $x$ is $0$ .

$\frac{x^{2} + 9 - x (2 x + 0)}{{(x^{2} + 9)}^{2}}$

Step 1.2.6

Simplify the expression.

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Step 1.2.6.1

Add $2 x$ and $0$ .

$\frac{x^{2} + 9 - x (2 x)}{{(x^{2} + 9)}^{2}}$

Step 1.2.6.2

Multiply $2$ by $- 1$ .

$\frac{x^{2} + 9 - 2 x \cdot x}{{(x^{2} + 9)}^{2}}$

Step 1.3

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

$\frac{x^{2} + 9 - 2 (x^{1} x)}{{(x^{2} + 9)}^{2}}$

Step 1.4

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

$\frac{x^{2} + 9 - 2 (x^{1} x^{1})}{{(x^{2} + 9)}^{2}}$

Step 1.5

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

$\frac{x^{2} + 9 - 2 x^{1 + 1}}{{(x^{2} + 9)}^{2}}$

Step 1.6

Add $1$ and $1$ .

$\frac{x^{2} + 9 - 2 x^{2}}{{(x^{2} + 9)}^{2}}$

Step 1.7

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

$\frac{- x^{2} + 9}{{(x^{2} + 9)}^{2}}$

Step 2

Set the derivative equal to $0$ then solve the equation $\frac{- x^{2} + 9}{{(x^{2} + 9)}^{2}} = 0$ .

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Step 2.1

Set the numerator equal to zero.

$- x^{2} + 9 = 0$

Step 2.2

Solve the equation for $x$ .

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Step 2.2.1

Subtract $9$ from both sides of the equation.

$- x^{2} = - 9$

Step 2.2.2

Divide each term in $- x^{2} = - 9$ by $- 1$ and simplify.

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Step 2.2.2.1

Divide each term in $- x^{2} = - 9$ by $- 1$ .

$\frac{- x^{2}}{- 1} = \frac{- 9}{- 1}$

Step 2.2.2.2

Simplify the left side.

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Step 2.2.2.2.1

Dividing two negative values results in a positive value.

$\frac{x^{2}}{1} = \frac{- 9}{- 1}$

Step 2.2.2.2.2

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

$x^{2} = \frac{- 9}{- 1}$

Step 2.2.2.3

Simplify the right side.

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Step 2.2.2.3.1

Divide $- 9$ by $- 1$ .

$x^{2} = 9$

Step 2.2.3

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

$x = \pm \sqrt{9}$

Step 2.2.4

Simplify $\pm \sqrt{9}$ .

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Step 2.2.4.1

Rewrite $9$ as $3^{2}$ .

$x = \pm \sqrt{3^{2}}$

Step 2.2.4.2

Pull terms out from under the radical, assuming positive real numbers.

$x = \pm 3$

Step 2.2.5

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

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Step 2.2.5.1

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

$x = 3$

Step 2.2.5.2

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

$x = - 3$

Step 2.2.5.3

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

$x = 3, - 3$

Step 3

Solve the original function $f (x) = \frac{x}{x^{2} + 9}$ at $x = 3$ .

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Step 3.1

Replace the variable $x$ with $3$ in the expression.

$f (3) = \frac{3}{{(3)}^{2} + 9}$

Step 3.2

Simplify the result.

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Step 3.2.1

Simplify the denominator.

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Step 3.2.1.1

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

$f (3) = \frac{3}{9 + 9}$

Step 3.2.1.2

Add $9$ and $9$ .

$f (3) = \frac{3}{18}$

Step 3.2.2

Cancel the common factor of $3$ and $18$ .

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Step 3.2.2.1

Factor $3$ out of $3$ .

$f (3) = \frac{3 (1)}{18}$

Step 3.2.2.2

Cancel the common factors.

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Step 3.2.2.2.1

Factor $3$ out of $18$ .

$f (3) = \frac{3 \cdot 1}{3 \cdot 6}$

Step 3.2.2.2.2

Cancel the common factor.

$f (3) = \frac{3 \cdot 1}{3 \cdot 6}$

Step 3.2.2.2.3

Rewrite the expression.

$f (3) = \frac{1}{6}$

Step 3.2.3

The final answer is $\frac{1}{6}$ .

$\frac{1}{6}$

Step 4

Solve the original function $f (x) = \frac{x}{x^{2} + 9}$ at $x = - 3$ .

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Step 4.1

Replace the variable $x$ with $- 3$ in the expression.

$f (- 3) = \frac{- 3}{{(- 3)}^{2} + 9}$

Step 4.2

Simplify the result.

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Step 4.2.1

Simplify the denominator.

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Step 4.2.1.1

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

$f (- 3) = \frac{- 3}{9 + 9}$

Step 4.2.1.2

Add $9$ and $9$ .

$f (- 3) = \frac{- 3}{18}$

Step 4.2.2

Reduce the expression by cancelling the common factors.

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Step 4.2.2.1

Cancel the common factor of $- 3$ and $18$ .

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Step 4.2.2.1.1

Factor $3$ out of $- 3$ .

$f (- 3) = \frac{3 (- 1)}{18}$

Step 4.2.2.1.2

Cancel the common factors.

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Step 4.2.2.1.2.1

Factor $3$ out of $18$ .

$f (- 3) = \frac{3 \cdot - 1}{3 \cdot 6}$

Step 4.2.2.1.2.2

Cancel the common factor.

$f (- 3) = \frac{3 \cdot - 1}{3 \cdot 6}$

Step 4.2.2.1.2.3

Rewrite the expression.

$f (- 3) = \frac{- 1}{6}$

Step 4.2.2.2

Move the negative in front of the fraction.

$f (- 3) = - \frac{1}{6}$

Step 4.2.3

The final answer is $- \frac{1}{6}$ .

$- \frac{1}{6}$

Step 5

The horizontal tangent lines on function $f (x) = \frac{x}{x^{2} + 9}$ are $y = \frac{1}{6}, y = - \frac{1}{6}$ .

$y = \frac{1}{6}, y = - \frac{1}{6}$

Step 6