Calculus Examples

$f (x) = \frac{100 x^{2} + 81}{x}$

Step 1

Find the first derivative.

Tap for more steps...

Step 1.1

Find the first derivative.

Tap for more steps...

Step 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 $f (x) = 100 x^{2} + 81$ and $g (x) = x$ .

$\frac{x \frac{d}{d x} [100 x^{2} + 81] - (100 x^{2} + 81) \frac{d}{d x} [x]}{x^{2}}$

Step 1.1.2

Differentiate.

Tap for more steps...

Step 1.1.2.1

By the Sum Rule, the derivative of $100 x^{2} + 81$ with respect to $x$ is $\frac{d}{d x} [100 x^{2}] + \frac{d}{d x} [81]$ .

$\frac{x (\frac{d}{d x} [100 x^{2}] + \frac{d}{d x} [81]) - (100 x^{2} + 81) \frac{d}{d x} [x]}{x^{2}}$

Step 1.1.2.2

Since $100$ is constant with respect to $x$ , the derivative of $100 x^{2}$ with respect to $x$ is $100 \frac{d}{d x} [x^{2}]$ .

$\frac{x (100 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [81]) - (100 x^{2} + 81) \frac{d}{d x} [x]}{x^{2}}$

Step 1.1.2.3

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

$\frac{x (100 (2 x) + \frac{d}{d x} [81]) - (100 x^{2} + 81) \frac{d}{d x} [x]}{x^{2}}$

Step 1.1.2.4

Multiply $2$ by $100$ .

$\frac{x (200 x + \frac{d}{d x} [81]) - (100 x^{2} + 81) \frac{d}{d x} [x]}{x^{2}}$

Step 1.1.2.5

Since $81$ is constant with respect to $x$ , the derivative of $81$ with respect to $x$ is $0$ .

$\frac{x (200 x + 0) - (100 x^{2} + 81) \frac{d}{d x} [x]}{x^{2}}$

Step 1.1.2.6

Add $200 x$ and $0$ .

$\frac{x (200 x) - (100 x^{2} + 81) \frac{d}{d x} [x]}{x^{2}}$

Step 1.1.3

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

$\frac{200 (x^{1} x) - (100 x^{2} + 81) \frac{d}{d x} [x]}{x^{2}}$

Step 1.1.4

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

$\frac{200 (x^{1} x^{1}) - (100 x^{2} + 81) \frac{d}{d x} [x]}{x^{2}}$

Step 1.1.5

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

$\frac{200 x^{1 + 1} - (100 x^{2} + 81) \frac{d}{d x} [x]}{x^{2}}$

Step 1.1.6

Add $1$ and $1$ .

$\frac{200 x^{2} - (100 x^{2} + 81) \frac{d}{d x} [x]}{x^{2}}$

Step 1.1.7

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

$\frac{200 x^{2} - (100 x^{2} + 81) \cdot 1}{x^{2}}$

Step 1.1.8

Multiply $- 1$ by $1$ .

$\frac{200 x^{2} - (100 x^{2} + 81)}{x^{2}}$

Step 1.1.9

Simplify.

Tap for more steps...

Step 1.1.9.1

Apply the distributive property.

$\frac{200 x^{2} - (100 x^{2}) - 1 \cdot 81}{x^{2}}$

Step 1.1.9.2

Simplify the numerator.

Tap for more steps...

Step 1.1.9.2.1

Simplify each term.

Tap for more steps...

Step 1.1.9.2.1.1

Multiply $100$ by $- 1$ .

$\frac{200 x^{2} - 100 x^{2} - 1 \cdot 81}{x^{2}}$

Step 1.1.9.2.1.2

Multiply $- 1$ by $81$ .

$\frac{200 x^{2} - 100 x^{2} - 81}{x^{2}}$

Step 1.1.9.2.2

Subtract $100 x^{2}$ from $200 x^{2}$ .

$\frac{100 x^{2} - 81}{x^{2}}$

Step 1.1.9.3

Simplify the numerator.

Tap for more steps...

Step 1.1.9.3.1

Rewrite $100 x^{2}$ as ${(10 x)}^{2}$ .

$\frac{{(10 x)}^{2} - 81}{x^{2}}$

Step 1.1.9.3.2

Rewrite $81$ as $9^{2}$ .

$\frac{{(10 x)}^{2} - 9^{2}}{x^{2}}$

Step 1.1.9.3.3

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 10 x$ and $b = 9$ .

$f' (x) = \frac{(10 x + 9) (10 x - 9)}{x^{2}}$

Step 1.2

The first derivative of $f (x)$ with respect to $x$ is $\frac{(10 x + 9) (10 x - 9)}{x^{2}}$ .

$\frac{(10 x + 9) (10 x - 9)}{x^{2}}$

Step 2

Set the first derivative equal to $0$ then solve the equation $\frac{(10 x + 9) (10 x - 9)}{x^{2}} = 0$ .

Tap for more steps...

Step 2.1

Set the first derivative equal to $0$ .

$\frac{(10 x + 9) (10 x - 9)}{x^{2}} = 0$

Step 2.2

Set the numerator equal to zero.

$(10 x + 9) (10 x - 9) = 0$

Step 2.3

Solve the equation for $x$ .

Tap for more steps...

Step 2.3.1

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$10 x + 9 = 0$

$10 x - 9 = 0$

Step 2.3.2

Set $10 x + 9$ equal to $0$ and solve for $x$ .

Tap for more steps...

Step 2.3.2.1

Set $10 x + 9$ equal to $0$ .

$10 x + 9 = 0$

Step 2.3.2.2

Solve $10 x + 9 = 0$ for $x$ .

Tap for more steps...

Step 2.3.2.2.1

Subtract $9$ from both sides of the equation.

$10 x = - 9$

Step 2.3.2.2.2

Divide each term in $10 x = - 9$ by $10$ and simplify.

Tap for more steps...

Step 2.3.2.2.2.1

Divide each term in $10 x = - 9$ by $10$ .

$\frac{10 x}{10} = \frac{- 9}{10}$

Step 2.3.2.2.2.2

Simplify the left side.

Tap for more steps...

Step 2.3.2.2.2.2.1

Cancel the common factor of $10$ .

Tap for more steps...

Step 2.3.2.2.2.2.1.1

Cancel the common factor.

$\frac{10 x}{10} = \frac{- 9}{10}$

Step 2.3.2.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 9}{10}$

Step 2.3.2.2.2.3

Simplify the right side.

Tap for more steps...

Step 2.3.2.2.2.3.1

Move the negative in front of the fraction.

$x = - \frac{9}{10}$

Step 2.3.3

Set $10 x - 9$ equal to $0$ and solve for $x$ .

Tap for more steps...

Step 2.3.3.1

Set $10 x - 9$ equal to $0$ .

$10 x - 9 = 0$

Step 2.3.3.2

Solve $10 x - 9 = 0$ for $x$ .

Tap for more steps...

Step 2.3.3.2.1

Add $9$ to both sides of the equation.

$10 x = 9$

Step 2.3.3.2.2

Divide each term in $10 x = 9$ by $10$ and simplify.

Tap for more steps...

Step 2.3.3.2.2.1

Divide each term in $10 x = 9$ by $10$ .

$\frac{10 x}{10} = \frac{9}{10}$

Step 2.3.3.2.2.2

Simplify the left side.

Tap for more steps...

Step 2.3.3.2.2.2.1

Cancel the common factor of $10$ .

Tap for more steps...

Step 2.3.3.2.2.2.1.1

Cancel the common factor.

$\frac{10 x}{10} = \frac{9}{10}$

Step 2.3.3.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{9}{10}$

Step 2.3.4

The final solution is all the values that make $(10 x + 9) (10 x - 9) = 0$ true.

$x = - \frac{9}{10}, \frac{9}{10}$

Step 3

Find the values where the derivative is undefined.

Tap for more steps...

Step 3.1

Set the denominator in $\frac{(10 x + 9) (10 x - 9)}{x^{2}}$ equal to $0$ to find where the expression is undefined.

$x^{2} = 0$

Step 3.2

Solve for $x$ .

Tap for more steps...

Step 3.2.1

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

$x = \pm \sqrt{0}$

Step 3.2.2

Simplify $\pm \sqrt{0}$ .

Tap for more steps...

Step 3.2.2.1

Rewrite $0$ as $0^{2}$ .

$x = \pm \sqrt{0^{2}}$

Step 3.2.2.2

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

$x = \pm 0$

Step 3.2.2.3

Plus or minus $0$ is $0$ .

$x = 0$

Step 4

Evaluate $\frac{100 x^{2} + 81}{x}$ at each $x$ value where the derivative is $0$ or undefined.

Tap for more steps...

Step 4.1

Evaluate at $x = - \frac{9}{10}$ .

Tap for more steps...

Step 4.1.1

Substitute $- \frac{9}{10}$ for $x$ .

$\frac{100 {(- \frac{9}{10})}^{2} + 81}{- \frac{9}{10}}$

Step 4.1.2

Simplify.

Tap for more steps...

Step 4.1.2.1

Multiply the numerator by the reciprocal of the denominator.

$(100 {(- \frac{9}{10})}^{2} + 81) (- \frac{10}{9})$

Step 4.1.2.2

Simplify each term.

Tap for more steps...

Step 4.1.2.2.1

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

Tap for more steps...

Step 4.1.2.2.1.1

Apply the product rule to $- \frac{9}{10}$ .

$(100 ({(- 1)}^{2} {(\frac{9}{10})}^{2}) + 81) (- \frac{10}{9})$

Step 4.1.2.2.1.2

Apply the product rule to $\frac{9}{10}$ .

$(100 ({(- 1)}^{2} \frac{9^{2}}{10^{2}}) + 81) (- \frac{10}{9})$

Step 4.1.2.2.2

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

$(100 (1 \frac{9^{2}}{10^{2}}) + 81) (- \frac{10}{9})$

Step 4.1.2.2.3

Multiply $\frac{9^{2}}{10^{2}}$ by $1$ .

$(100 \frac{9^{2}}{10^{2}} + 81) (- \frac{10}{9})$

Step 4.1.2.2.4

Raise $9$ to the power of $2$ .

$(100 \frac{81}{10^{2}} + 81) (- \frac{10}{9})$

Step 4.1.2.2.5

Raise $10$ to the power of $2$ .

$(100 (\frac{81}{100}) + 81) (- \frac{10}{9})$

Step 4.1.2.2.6

Cancel the common factor of $100$ .

Tap for more steps...

Step 4.1.2.2.6.1

Cancel the common factor.

$(100 (\frac{81}{100}) + 81) (- \frac{10}{9})$

Step 4.1.2.2.6.2

Rewrite the expression.

$(81 + 81) (- \frac{10}{9})$

Step 4.1.2.3

Reduce the expression by cancelling the common factors.

Tap for more steps...

Step 4.1.2.3.1

Add $81$ and $81$ .

$162 (- \frac{10}{9})$

Step 4.1.2.3.2

Cancel the common factor of $9$ .

Tap for more steps...

Step 4.1.2.3.2.1

Move the leading negative in $- \frac{10}{9}$ into the numerator.

$162 (\frac{- 10}{9})$

Step 4.1.2.3.2.2

Factor $9$ out of $162$ .

$9 (18) \frac{- 10}{9}$

Step 4.1.2.3.2.3

Cancel the common factor.

$9 \cdot 18 \frac{- 10}{9}$

Step 4.1.2.3.2.4

Rewrite the expression.

$18 \cdot - 10$

Step 4.1.2.3.3

Multiply $18$ by $- 10$ .

$- 180$

Step 4.2

Evaluate at $x = \frac{9}{10}$ .

Tap for more steps...

Step 4.2.1

Substitute $\frac{9}{10}$ for $x$ .

$\frac{100 {(\frac{9}{10})}^{2} + 81}{\frac{9}{10}}$

Step 4.2.2

Simplify.

Tap for more steps...

Step 4.2.2.1

Multiply the numerator by the reciprocal of the denominator.

$(100 {(\frac{9}{10})}^{2} + 81) \frac{10}{9}$

Step 4.2.2.2

Simplify each term.

Tap for more steps...

Step 4.2.2.2.1

Apply the product rule to $\frac{9}{10}$ .

$(100 \frac{9^{2}}{10^{2}} + 81) \frac{10}{9}$

Step 4.2.2.2.2

Raise $9$ to the power of $2$ .

$(100 \frac{81}{10^{2}} + 81) \frac{10}{9}$

Step 4.2.2.2.3

Raise $10$ to the power of $2$ .

$(100 (\frac{81}{100}) + 81) \frac{10}{9}$

Step 4.2.2.2.4

Cancel the common factor of $100$ .

Tap for more steps...

Step 4.2.2.2.4.1

Cancel the common factor.

$(100 (\frac{81}{100}) + 81) \frac{10}{9}$

Step 4.2.2.2.4.2

Rewrite the expression.

$(81 + 81) \frac{10}{9}$

Step 4.2.2.3

Reduce the expression by cancelling the common factors.

Tap for more steps...

Step 4.2.2.3.1

Add $81$ and $81$ .

$162 (\frac{10}{9})$

Step 4.2.2.3.2

Cancel the common factor of $9$ .

Tap for more steps...

Step 4.2.2.3.2.1

Factor $9$ out of $162$ .

$9 (18) \frac{10}{9}$

Step 4.2.2.3.2.2

Cancel the common factor.

$9 \cdot 18 \frac{10}{9}$

Step 4.2.2.3.2.3

Rewrite the expression.

$18 \cdot 10$

Step 4.2.2.3.3

Multiply $18$ by $10$ .

$180$

Step 4.3

Evaluate at $x = 0$ .

Tap for more steps...

Step 4.3.1

Substitute $0$ for $x$ .

$\frac{100 {(0)}^{2} + 81}{0}$

Step 4.3.2

The expression contains a division by $0$ . The expression is undefined.

Undefined

Step 4.4

List all of the points.

$(- \frac{9}{10}, - 180), (\frac{9}{10}, 180)$

Step 5