Calculus Examples

$f (x) = 2 x^{2} + \frac{108}{x}$

Step 1

Find the first derivative.

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Step 1.1

Find the first derivative.

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Step 1.1.1

By the Sum Rule, the derivative of $2 x^{2} + \frac{108}{x}$ with respect to $x$ is $\frac{d}{d x} [2 x^{2}] + \frac{d}{d x} [\frac{108}{x}]$ .

$f' (x) = \frac{d}{d x} (2 x^{2}) + \frac{d}{d x} (\frac{108}{x})$

Step 1.1.2

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

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Step 1.1.2.1

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

$f' (x) = 2 \frac{d}{d x} (x^{2}) + \frac{d}{d x} (\frac{108}{x})$

Step 1.1.2.2

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

$f' (x) = 2 (2 x) + \frac{d}{d x} (\frac{108}{x})$

Step 1.1.2.3

Multiply $2$ by $2$ .

$f' (x) = 4 x + \frac{d}{d x} (\frac{108}{x})$

Step 1.1.3

Evaluate $\frac{d}{d x} [\frac{108}{x}]$ .

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Step 1.1.3.1

Since $108$ is constant with respect to $x$ , the derivative of $\frac{108}{x}$ with respect to $x$ is $108 \frac{d}{d x} [\frac{1}{x}]$ .

$f' (x) = 4 x + 108 \frac{d}{d x} (\frac{1}{x})$

Step 1.1.3.2

Rewrite $\frac{1}{x}$ as $x^{- 1}$ .

$f' (x) = 4 x + 108 \frac{d}{d x} (x^{- 1})$

Step 1.1.3.3

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

$f' (x) = 4 x + 108 (- x^{- 2})$

Step 1.1.3.4

Multiply $- 1$ by $108$ .

$f' (x) = 4 x - 108 x^{- 2}$

Step 1.1.4

Simplify.

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Step 1.1.4.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$f' (x) = 4 x - 108 \frac{1}{x^{2}}$

Step 1.1.4.2

Combine terms.

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Step 1.1.4.2.1

Combine $- 108$ and $\frac{1}{x^{2}}$ .

$f' (x) = 4 x + \frac{- 108}{x^{2}}$

Step 1.1.4.2.2

Move the negative in front of the fraction.

$f' (x) = 4 x - \frac{108}{x^{2}}$

Step 1.2

The first derivative of $f (x)$ with respect to $x$ is $4 x - \frac{108}{x^{2}}$ .

$4 x - \frac{108}{x^{2}}$

Step 2

Set the first derivative equal to $0$ then solve the equation $4 x - \frac{108}{x^{2}} = 0$ .

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Step 2.1

Set the first derivative equal to $0$ .

$4 x - \frac{108}{x^{2}} = 0$

Step 2.2

Find the LCD of the terms in the equation.

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Step 2.2.1

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$1, x^{2}, 1$

Step 2.2.2

The LCM of one and any expression is the expression.

$x^{2}$

Step 2.3

Multiply each term in $4 x - \frac{108}{x^{2}} = 0$ by $x^{2}$ to eliminate the fractions.

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Step 2.3.1

Multiply each term in $4 x - \frac{108}{x^{2}} = 0$ by $x^{2}$ .

$4 x \cdot x^{2} - \frac{108}{x^{2}} x^{2} = 0 x^{2}$

Step 2.3.2

Simplify the left side.

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Step 2.3.2.1

Simplify each term.

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Step 2.3.2.1.1

Multiply $x$ by $x^{2}$ by adding the exponents.

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Step 2.3.2.1.1.1

Move $x^{2}$ .

$4 (x^{2} x) - \frac{108}{x^{2}} x^{2} = 0 x^{2}$

Step 2.3.2.1.1.2

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

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Step 2.3.2.1.1.2.1

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

$4 (x^{2} x^{1}) - \frac{108}{x^{2}} x^{2} = 0 x^{2}$

Step 2.3.2.1.1.2.2

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

$4 x^{2 + 1} - \frac{108}{x^{2}} x^{2} = 0 x^{2}$

Step 2.3.2.1.1.3

Add $2$ and $1$ .

$4 x^{3} - \frac{108}{x^{2}} x^{2} = 0 x^{2}$

Step 2.3.2.1.2

Cancel the common factor of $x^{2}$ .

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Step 2.3.2.1.2.1

Move the leading negative in $- \frac{108}{x^{2}}$ into the numerator.

$4 x^{3} + \frac{- 108}{x^{2}} x^{2} = 0 x^{2}$

Step 2.3.2.1.2.2

Cancel the common factor.

$4 x^{3} + \frac{- 108}{x^{2}} x^{2} = 0 x^{2}$

Step 2.3.2.1.2.3

Rewrite the expression.

$4 x^{3} - 108 = 0 x^{2}$

Step 2.3.3

Simplify the right side.

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Step 2.3.3.1

Multiply $0$ by $x^{2}$ .

$4 x^{3} - 108 = 0$

Step 2.4

Solve the equation.

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Step 2.4.1

Add $108$ to both sides of the equation.

$4 x^{3} = 108$

Step 2.4.2

Subtract $108$ from both sides of the equation.

$4 x^{3} - 108 = 0$

Step 2.4.3

Factor the left side of the equation.

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Step 2.4.3.1

Factor $4$ out of $4 x^{3} - 108$ .

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Step 2.4.3.1.1

Factor $4$ out of $4 x^{3}$ .

$4 (x^{3}) - 108 = 0$

Step 2.4.3.1.2

Factor $4$ out of $- 108$ .

$4 x^{3} + 4 \cdot - 27 = 0$

Step 2.4.3.1.3

Factor $4$ out of $4 x^{3} + 4 \cdot - 27$ .

$4 (x^{3} - 27) = 0$

Step 2.4.3.2

Rewrite $27$ as $3^{3}$ .

$4 (x^{3} - 3^{3}) = 0$

Step 2.4.3.3

Since both terms are perfect cubes, factor using the difference of cubes formula, $a^{3} - b^{3} = (a - b) (a^{2} + a b + b^{2})$ where $a = x$ and $b = 3$ .

$4 ((x - 3) (x^{2} + x \cdot 3 + 3^{2})) = 0$

Step 2.4.3.4

Factor.

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Step 2.4.3.4.1

Simplify.

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Step 2.4.3.4.1.1

Move $3$ to the left of $x$ .

$4 ((x - 3) (x^{2} + 3 x + 3^{2})) = 0$

Step 2.4.3.4.1.2

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

$4 ((x - 3) (x^{2} + 3 x + 9)) = 0$

Step 2.4.3.4.2

Remove unnecessary parentheses.

$4 (x - 3) (x^{2} + 3 x + 9) = 0$

Step 2.4.4

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

$x - 3 = 0$

$x^{2} + 3 x + 9 = 0$

Step 2.4.5

Set $x - 3$ equal to $0$ and solve for $x$ .

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Step 2.4.5.1

Set $x - 3$ equal to $0$ .

$x - 3 = 0$

Step 2.4.5.2

Add $3$ to both sides of the equation.

$x = 3$

Step 2.4.6

Set $x^{2} + 3 x + 9$ equal to $0$ and solve for $x$ .

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Step 2.4.6.1

Set $x^{2} + 3 x + 9$ equal to $0$ .

$x^{2} + 3 x + 9 = 0$

Step 2.4.6.2

Solve $x^{2} + 3 x + 9 = 0$ for $x$ .

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Step 2.4.6.2.1

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 2.4.6.2.2

Substitute the values $a = 1$ , $b = 3$ , and $c = 9$ into the quadratic formula and solve for $x$ .

$\frac{- 3 \pm \sqrt{3^{2} - 4 \cdot (1 \cdot 9)}}{2 \cdot 1}$

Step 2.4.6.2.3

Simplify.

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Step 2.4.6.2.3.1

Simplify the numerator.

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Step 2.4.6.2.3.1.1

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

$x = \frac{- 3 \pm \sqrt{9 - 4 \cdot 1 \cdot 9}}{2 \cdot 1}$

Step 2.4.6.2.3.1.2

Multiply $- 4 \cdot 1 \cdot 9$ .

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Step 2.4.6.2.3.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{- 3 \pm \sqrt{9 - 4 \cdot 9}}{2 \cdot 1}$

Step 2.4.6.2.3.1.2.2

Multiply $- 4$ by $9$ .

$x = \frac{- 3 \pm \sqrt{9 - 36}}{2 \cdot 1}$

Step 2.4.6.2.3.1.3

Subtract $36$ from $9$ .

$x = \frac{- 3 \pm \sqrt{- 27}}{2 \cdot 1}$

Step 2.4.6.2.3.1.4

Rewrite $- 27$ as $- 1 (27)$ .

$x = \frac{- 3 \pm \sqrt{- 1 \cdot 27}}{2 \cdot 1}$

Step 2.4.6.2.3.1.5

Rewrite $\sqrt{- 1 (27)}$ as $\sqrt{- 1} \cdot \sqrt{27}$ .

$x = \frac{- 3 \pm \sqrt{- 1} \cdot \sqrt{27}}{2 \cdot 1}$

Step 2.4.6.2.3.1.6

Rewrite $\sqrt{- 1}$ as $i$ .

$x = \frac{- 3 \pm i \cdot \sqrt{27}}{2 \cdot 1}$

Step 2.4.6.2.3.1.7

Rewrite $27$ as $3^{2} \cdot 3$ .

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Step 2.4.6.2.3.1.7.1

Factor $9$ out of $27$ .

$x = \frac{- 3 \pm i \cdot \sqrt{9 (3)}}{2 \cdot 1}$

Step 2.4.6.2.3.1.7.2

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

$x = \frac{- 3 \pm i \cdot \sqrt{3^{2} \cdot 3}}{2 \cdot 1}$

Step 2.4.6.2.3.1.8

Pull terms out from under the radical.

$x = \frac{- 3 \pm i \cdot (3 \sqrt{3})}{2 \cdot 1}$

Step 2.4.6.2.3.1.9

Move $3$ to the left of $i$ .

$x = \frac{- 3 \pm 3 i \sqrt{3}}{2 \cdot 1}$

Step 2.4.6.2.3.2

Multiply $2$ by $1$ .

$x = \frac{- 3 \pm 3 i \sqrt{3}}{2}$

Step 2.4.6.2.4

Simplify the expression to solve for the $+$ portion of the $\pm$ .

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Step 2.4.6.2.4.1

Simplify the numerator.

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Step 2.4.6.2.4.1.1

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

$x = \frac{- 3 \pm \sqrt{9 - 4 \cdot 1 \cdot 9}}{2 \cdot 1}$

Step 2.4.6.2.4.1.2

Multiply $- 4 \cdot 1 \cdot 9$ .

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Step 2.4.6.2.4.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{- 3 \pm \sqrt{9 - 4 \cdot 9}}{2 \cdot 1}$

Step 2.4.6.2.4.1.2.2

Multiply $- 4$ by $9$ .

$x = \frac{- 3 \pm \sqrt{9 - 36}}{2 \cdot 1}$

Step 2.4.6.2.4.1.3

Subtract $36$ from $9$ .

$x = \frac{- 3 \pm \sqrt{- 27}}{2 \cdot 1}$

Step 2.4.6.2.4.1.4

Rewrite $- 27$ as $- 1 (27)$ .

$x = \frac{- 3 \pm \sqrt{- 1 \cdot 27}}{2 \cdot 1}$

Step 2.4.6.2.4.1.5

Rewrite $\sqrt{- 1 (27)}$ as $\sqrt{- 1} \cdot \sqrt{27}$ .

$x = \frac{- 3 \pm \sqrt{- 1} \cdot \sqrt{27}}{2 \cdot 1}$

Step 2.4.6.2.4.1.6

Rewrite $\sqrt{- 1}$ as $i$ .

$x = \frac{- 3 \pm i \cdot \sqrt{27}}{2 \cdot 1}$

Step 2.4.6.2.4.1.7

Rewrite $27$ as $3^{2} \cdot 3$ .

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Step 2.4.6.2.4.1.7.1

Factor $9$ out of $27$ .

$x = \frac{- 3 \pm i \cdot \sqrt{9 (3)}}{2 \cdot 1}$

Step 2.4.6.2.4.1.7.2

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

$x = \frac{- 3 \pm i \cdot \sqrt{3^{2} \cdot 3}}{2 \cdot 1}$

Step 2.4.6.2.4.1.8

Pull terms out from under the radical.

$x = \frac{- 3 \pm i \cdot (3 \sqrt{3})}{2 \cdot 1}$

Step 2.4.6.2.4.1.9

Move $3$ to the left of $i$ .

$x = \frac{- 3 \pm 3 i \sqrt{3}}{2 \cdot 1}$

Step 2.4.6.2.4.2

Multiply $2$ by $1$ .

$x = \frac{- 3 \pm 3 i \sqrt{3}}{2}$

Step 2.4.6.2.4.3

Change the $\pm$ to $+$ .

$x = \frac{- 3 + 3 i \sqrt{3}}{2}$

Step 2.4.6.2.4.4

Rewrite $- 3$ as $- 1 (3)$ .

$x = \frac{- 1 \cdot 3 + 3 i \sqrt{3}}{2}$

Step 2.4.6.2.4.5

Factor $- 1$ out of $3 i \sqrt{3}$ .

$x = \frac{- 1 \cdot 3 - (- 3 i \sqrt{3})}{2}$

Step 2.4.6.2.4.6

Factor $- 1$ out of $- 1 (3) - (- 3 i \sqrt{3})$ .

$x = \frac{- 1 (3 - 3 i \sqrt{3})}{2}$

Step 2.4.6.2.4.7

Move the negative in front of the fraction.

$x = - \frac{3 - 3 i \sqrt{3}}{2}$

Step 2.4.6.2.5

Simplify the expression to solve for the $-$ portion of the $\pm$ .

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Step 2.4.6.2.5.1

Simplify the numerator.

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Step 2.4.6.2.5.1.1

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

$x = \frac{- 3 \pm \sqrt{9 - 4 \cdot 1 \cdot 9}}{2 \cdot 1}$

Step 2.4.6.2.5.1.2

Multiply $- 4 \cdot 1 \cdot 9$ .

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Step 2.4.6.2.5.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{- 3 \pm \sqrt{9 - 4 \cdot 9}}{2 \cdot 1}$

Step 2.4.6.2.5.1.2.2

Multiply $- 4$ by $9$ .

$x = \frac{- 3 \pm \sqrt{9 - 36}}{2 \cdot 1}$

Step 2.4.6.2.5.1.3

Subtract $36$ from $9$ .

$x = \frac{- 3 \pm \sqrt{- 27}}{2 \cdot 1}$

Step 2.4.6.2.5.1.4

Rewrite $- 27$ as $- 1 (27)$ .

$x = \frac{- 3 \pm \sqrt{- 1 \cdot 27}}{2 \cdot 1}$

Step 2.4.6.2.5.1.5

Rewrite $\sqrt{- 1 (27)}$ as $\sqrt{- 1} \cdot \sqrt{27}$ .

$x = \frac{- 3 \pm \sqrt{- 1} \cdot \sqrt{27}}{2 \cdot 1}$

Step 2.4.6.2.5.1.6

Rewrite $\sqrt{- 1}$ as $i$ .

$x = \frac{- 3 \pm i \cdot \sqrt{27}}{2 \cdot 1}$

Step 2.4.6.2.5.1.7

Rewrite $27$ as $3^{2} \cdot 3$ .

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Step 2.4.6.2.5.1.7.1

Factor $9$ out of $27$ .

$x = \frac{- 3 \pm i \cdot \sqrt{9 (3)}}{2 \cdot 1}$

Step 2.4.6.2.5.1.7.2

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

$x = \frac{- 3 \pm i \cdot \sqrt{3^{2} \cdot 3}}{2 \cdot 1}$

Step 2.4.6.2.5.1.8

Pull terms out from under the radical.

$x = \frac{- 3 \pm i \cdot (3 \sqrt{3})}{2 \cdot 1}$

Step 2.4.6.2.5.1.9

Move $3$ to the left of $i$ .

$x = \frac{- 3 \pm 3 i \sqrt{3}}{2 \cdot 1}$

Step 2.4.6.2.5.2

Multiply $2$ by $1$ .

$x = \frac{- 3 \pm 3 i \sqrt{3}}{2}$

Step 2.4.6.2.5.3

Change the $\pm$ to $-$ .

$x = \frac{- 3 - 3 i \sqrt{3}}{2}$

Step 2.4.6.2.5.4

Rewrite $- 3$ as $- 1 (3)$ .

$x = \frac{- 1 \cdot 3 - 3 i \sqrt{3}}{2}$

Step 2.4.6.2.5.5

Factor $- 1$ out of $- 3 i \sqrt{3}$ .

$x = \frac{- 1 \cdot 3 - (3 i \sqrt{3})}{2}$

Step 2.4.6.2.5.6

Factor $- 1$ out of $- 1 (3) - (3 i \sqrt{3})$ .

$x = \frac{- 1 (3 + 3 i \sqrt{3})}{2}$

Step 2.4.6.2.5.7

Move the negative in front of the fraction.

$x = - \frac{3 + 3 i \sqrt{3}}{2}$

Step 2.4.6.2.6

The final answer is the combination of both solutions.

$x = - \frac{3 - 3 i \sqrt{3}}{2}, - \frac{3 + 3 i \sqrt{3}}{2}$

Step 2.4.7

The final solution is all the values that make $4 (x - 3) (x^{2} + 3 x + 9) = 0$ true.

$x = 3, - \frac{3 - 3 i \sqrt{3}}{2}, - \frac{3 + 3 i \sqrt{3}}{2}$

Step 3

Find the values where the derivative is undefined.

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Step 3.1

Set the denominator in $\frac{108}{x^{2}}$ equal to $0$ to find where the expression is undefined.

$x^{2} = 0$

Step 3.2

Solve for $x$ .

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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 $\pm \sqrt{0}$ .

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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 $2 x^{2} + \frac{108}{x}$ at each $x$ value where the derivative is $0$ or undefined.

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Step 4.1

Evaluate at $x = 3$ .

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Step 4.1.1

Substitute $3$ for $x$ .

$2 {(3)}^{2} + \frac{108}{3}$

Step 4.1.2

Simplify.

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Step 4.1.2.1

Simplify each term.

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Step 4.1.2.1.1

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

$2 \cdot 9 + \frac{108}{3}$

Step 4.1.2.1.2

Multiply $2$ by $9$ .

$18 + \frac{108}{3}$

Step 4.1.2.1.3

Divide $108$ by $3$ .

$18 + 36$

Step 4.1.2.2

Add $18$ and $36$ .

$54$

Step 4.2

Evaluate at $x = 0$ .

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Step 4.2.1

Substitute $0$ for $x$ .

$2 {(0)}^{2} + \frac{108}{0}$

Step 4.2.2

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

Undefined

Step 4.3

List all of the points.

$(3, 54)$

Step 5