Calculus Examples

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

Step 1

Find the first derivative of the function.

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Step 1.1

By the Sum Rule, the derivative of $4 x + \frac{1}{x}$ with respect to $x$ is $\frac{d}{d x} [4 x] + \frac{d}{d x} [\frac{1}{x}]$ .

$\frac{d}{d x} [4 x] + \frac{d}{d x} [\frac{1}{x}]$

Step 1.2

Evaluate $\frac{d}{d x} [4 x]$ .

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Step 1.2.1

Since $4$ is constant with respect to $x$ , the derivative of $4 x$ with respect to $x$ is $4 \frac{d}{d x} [x]$ .

$4 \frac{d}{d x} [x] + \frac{d}{d x} [\frac{1}{x}]$

Step 1.2.2

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

$4 \cdot 1 + \frac{d}{d x} [\frac{1}{x}]$

Step 1.2.3

Multiply $4$ by $1$ .

$4 + \frac{d}{d x} [\frac{1}{x}]$

Step 1.3

Evaluate $\frac{d}{d x} [\frac{1}{x}]$ .

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Step 1.3.1

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

$4 + \frac{d}{d x} [x^{- 1}]$

Step 1.3.2

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

$4 - x^{- 2}$

Step 1.4

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

$4 - \frac{1}{x^{2}}$

Step 1.5

Reorder terms.

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

Step 2

Find the second derivative of the function.

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Step 2.1

By the Sum Rule, the derivative of $- \frac{1}{x^{2}} + 4$ with respect to $x$ is $\frac{d}{d x} [- \frac{1}{x^{2}}] + \frac{d}{d x} [4]$ .

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

Step 2.2

Evaluate $\frac{d}{d x} [- \frac{1}{x^{2}}]$ .

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Step 2.2.1

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = - 1$ and $g (x) = \frac{1}{x^{2}}$ .

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

Step 2.2.2

Rewrite $\frac{1}{x^{2}}$ as ${(x^{2})}^{- 1}$ .

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

Step 2.2.3

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{- 1}$ and $g (x) = x^{2}$ .

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Step 2.2.3.1

To apply the Chain Rule, set $u$ as $x^{2}$ .

$f'' (x) = - (\frac{d}{d u} (u^{- 1}) \frac{d}{d x} (x^{2})) + \frac{1}{x^{2}} \cdot \frac{d}{d x} (- 1) + \frac{d}{d x} (4)$

Step 2.2.3.2

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

$f'' (x) = - (- u^{- 2} \frac{d}{d x} x^{2}) + \frac{1}{x^{2}} \cdot \frac{d}{d x} (- 1) + \frac{d}{d x} (4)$

Step 2.2.3.3

Replace all occurrences of $u$ with $x^{2}$ .

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

Step 2.2.4

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

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

Step 2.2.5

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

$f'' (x) = - (- {(x^{2})}^{- 2} (2 x)) + \frac{1}{x^{2}} \cdot 0 + \frac{d}{d x} (4)$

Step 2.2.6

Multiply the exponents in ${(x^{2})}^{- 2}$ .

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Step 2.2.6.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$f'' (x) = - (- x^{2 \cdot - 2} (2 x)) + \frac{1}{x^{2}} \cdot 0 + \frac{d}{d x} (4)$

Step 2.2.6.2

Multiply $2$ by $- 2$ .

$f'' (x) = - (- x^{- 4} (2 x)) + \frac{1}{x^{2}} \cdot 0 + \frac{d}{d x} (4)$

Step 2.2.7

Multiply $2$ by $- 1$ .

$f'' (x) = - (- 2 x^{- 4} x) + \frac{1}{x^{2}} \cdot 0 + \frac{d}{d x} (4)$

Step 2.2.8

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

$f'' (x) = - (- 2 (x \cdot x^{- 4})) + \frac{1}{x^{2}} \cdot 0 + \frac{d}{d x} (4)$

Step 2.2.9

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

$f'' (x) = - (- 2 x^{1 - 4}) + \frac{1}{x^{2}} \cdot 0 + \frac{d}{d x} (4)$

Step 2.2.10

Subtract $4$ from $1$ .

$f'' (x) = - (- 2 x^{- 3}) + \frac{1}{x^{2}} \cdot 0 + \frac{d}{d x} (4)$

Step 2.2.11

Multiply $- 2$ by $- 1$ .

$f'' (x) = 2 x^{- 3} + \frac{1}{x^{2}} \cdot 0 + \frac{d}{d x} (4)$

Step 2.2.12

Multiply $\frac{1}{x^{2}}$ by $0$ .

$f'' (x) = 2 x^{- 3} + 0 + \frac{d}{d x} (4)$

Step 2.2.13

Add $2 x^{- 3}$ and $0$ .

$f'' (x) = 2 x^{- 3} + \frac{d}{d x} (4)$

Step 2.3

Since $4$ is constant with respect to $x$ , the derivative of $4$ with respect to $x$ is $0$ .

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

Step 2.4

Simplify.

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Step 2.4.1

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

$f'' (x) = 2 (\frac{1}{x^{3}}) + 0$

Step 2.4.2

Combine terms.

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Step 2.4.2.1

Combine $2$ and $\frac{1}{x^{3}}$ .

$f'' (x) = \frac{2}{x^{3}} + 0$

Step 2.4.2.2

Add $\frac{2}{x^{3}}$ and $0$ .

$f'' (x) = \frac{2}{x^{3}}$

Step 3

To find the local maximum and minimum values of the function, set the derivative equal to $0$ and solve.

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

Step 4

Find the first derivative.

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Step 4.1

Find the first derivative.

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Step 4.1.1

By the Sum Rule, the derivative of $4 x + \frac{1}{x}$ with respect to $x$ is $\frac{d}{d x} [4 x] + \frac{d}{d x} [\frac{1}{x}]$ .

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

Step 4.1.2

Evaluate $\frac{d}{d x} [4 x]$ .

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Step 4.1.2.1

Since $4$ is constant with respect to $x$ , the derivative of $4 x$ with respect to $x$ is $4 \frac{d}{d x} [x]$ .

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

Step 4.1.2.2

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

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

Step 4.1.2.3

Multiply $4$ by $1$ .

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

Step 4.1.3

Evaluate $\frac{d}{d x} [\frac{1}{x}]$ .

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Step 4.1.3.1

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

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

Step 4.1.3.2

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

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

Step 4.1.4

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

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

Step 4.1.5

Reorder terms.

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

Step 4.2

The first derivative of $f (x)$ with respect to $x$ is $- \frac{1}{x^{2}} + 4$ .

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

Step 5

Set the first derivative equal to $0$ then solve the equation $- \frac{1}{x^{2}} + 4 = 0$ .

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Step 5.1

Set the first derivative equal to $0$ .

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

Step 5.2

Subtract $4$ from both sides of the equation.

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

Step 5.3

Find the LCD of the terms in the equation.

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Step 5.3.1

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

$x^{2}, 1$

Step 5.3.2

The LCM of one and any expression is the expression.

$x^{2}$

Step 5.4

Multiply each term in $- \frac{1}{x^{2}} = - 4$ by $x^{2}$ to eliminate the fractions.

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Step 5.4.1

Multiply each term in $- \frac{1}{x^{2}} = - 4$ by $x^{2}$ .

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

Step 5.4.2

Simplify the left side.

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Step 5.4.2.1

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

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Step 5.4.2.1.1

Move the leading negative in $- \frac{1}{x^{2}}$ into the numerator.

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

Step 5.4.2.1.2

Cancel the common factor.

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

Step 5.4.2.1.3

Rewrite the expression.

$- 1 = - 4 x^{2}$

Step 5.5

Solve the equation.

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Step 5.5.1

Rewrite the equation as $- 4 x^{2} = - 1$ .

$- 4 x^{2} = - 1$

Step 5.5.2

Divide each term in $- 4 x^{2} = - 1$ by $- 4$ and simplify.

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Step 5.5.2.1

Divide each term in $- 4 x^{2} = - 1$ by $- 4$ .

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

Step 5.5.2.2

Simplify the left side.

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Step 5.5.2.2.1

Cancel the common factor of $- 4$ .

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Step 5.5.2.2.1.1

Cancel the common factor.

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

Step 5.5.2.2.1.2

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

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

Step 5.5.2.3

Simplify the right side.

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Step 5.5.2.3.1

Dividing two negative values results in a positive value.

$x^{2} = \frac{1}{4}$

Step 5.5.3

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

$x = \pm \sqrt{\frac{1}{4}}$

Step 5.5.4

Simplify $\pm \sqrt{\frac{1}{4}}$ .

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Step 5.5.4.1

Rewrite $\sqrt{\frac{1}{4}}$ as $\frac{\sqrt{1}}{\sqrt{4}}$ .

$x = \pm \frac{\sqrt{1}}{\sqrt{4}}$

Step 5.5.4.2

Any root of $1$ is $1$ .

$x = \pm \frac{1}{\sqrt{4}}$

Step 5.5.4.3

Simplify the denominator.

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Step 5.5.4.3.1

Rewrite $4$ as $2^{2}$ .

$x = \pm \frac{1}{\sqrt{2^{2}}}$

Step 5.5.4.3.2

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

$x = \pm \frac{1}{2}$

Step 5.5.5

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

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Step 5.5.5.1

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

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

Step 5.5.5.2

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

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

Step 5.5.5.3

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

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

Step 6

Find the values where the derivative is undefined.

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Step 6.1

Set the denominator in $\frac{1}{x^{2}}$ equal to $0$ to find where the expression is undefined.

$x^{2} = 0$

Step 6.2

Solve for $x$ .

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Step 6.2.1

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

$x = \pm \sqrt{0}$

Step 6.2.2

Simplify $\pm \sqrt{0}$ .

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Step 6.2.2.1

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

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

Step 6.2.2.2

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

$x = \pm 0$

Step 6.2.2.3

Plus or minus $0$ is $0$ .

$x = 0$

Step 7

Critical points to evaluate.

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

Step 8

Evaluate the second derivative at $x = \frac{1}{2}$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

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

Step 9

Evaluate the second derivative.

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Step 9.1

Simplify the denominator.

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Step 9.1.1

Apply the product rule to $\frac{1}{2}$ .

$\frac{2}{\frac{1^{3}}{2^{3}}}$

Step 9.1.2

One to any power is one.

$\frac{2}{\frac{1}{2^{3}}}$

Step 9.1.3

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

$\frac{2}{\frac{1}{8}}$

Step 9.2

Multiply the numerator by the reciprocal of the denominator.

$2 \cdot 8$

Step 9.3

Multiply $2$ by $8$ .

$16$

Step 10

$x = \frac{1}{2}$ is a local minimum because the value of the second derivative is positive. This is referred to as the second derivative test.

$x = \frac{1}{2}$ is a local minimum

Step 11

Find the y-value when $x = \frac{1}{2}$ .

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Step 11.1

Replace the variable $x$ with $\frac{1}{2}$ in the expression.

$f (\frac{1}{2}) = 4 (\frac{1}{2}) + \frac{1}{\frac{1}{2}}$

Step 11.2

Simplify the result.

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Step 11.2.1

Simplify each term.

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Step 11.2.1.1

Cancel the common factor of $2$ .

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Step 11.2.1.1.1

Factor $2$ out of $4$ .

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

Step 11.2.1.1.2

Cancel the common factor.

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

Step 11.2.1.1.3

Rewrite the expression.

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

Step 11.2.1.2

Multiply the numerator by the reciprocal of the denominator.

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

Step 11.2.1.3

Multiply $2$ by $1$ .

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

Step 11.2.2

Add $2$ and $2$ .

$f (\frac{1}{2}) = 4$

Step 11.2.3

The final answer is $4$ .

$y = 4$

Step 12

Evaluate the second derivative at $x = - \frac{1}{2}$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

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

Step 13

Evaluate the second derivative.

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Step 13.1

Simplify the denominator.

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Step 13.1.1

Apply the product rule to $- \frac{1}{2}$ .

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

Step 13.1.2

Raise $- 1$ to the power of $3$ .

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

Step 13.1.3

Apply the product rule to $\frac{1}{2}$ .

$\frac{2}{- \frac{1^{3}}{2^{3}}}$

Step 13.1.4

One to any power is one.

$\frac{2}{- \frac{1}{2^{3}}}$

Step 13.1.5

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

$\frac{2}{- \frac{1}{8}}$

Step 13.2

Multiply the numerator by the reciprocal of the denominator.

$2 (- 1 \cdot 8)$

Step 13.3

Multiply $2 (- 1 \cdot 8)$ .

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Step 13.3.1

Multiply $- 1$ by $8$ .

$2 \cdot - 8$

Step 13.3.2

Multiply $2$ by $- 8$ .

$- 16$

Step 14

$x = - \frac{1}{2}$ is a local maximum because the value of the second derivative is negative. This is referred to as the second derivative test.

$x = - \frac{1}{2}$ is a local maximum

Step 15

Find the y-value when $x = - \frac{1}{2}$ .

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Step 15.1

Replace the variable $x$ with $- \frac{1}{2}$ in the expression.

$f (- \frac{1}{2}) = 4 (- \frac{1}{2}) + \frac{1}{- \frac{1}{2}}$

Step 15.2

Simplify the result.

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Step 15.2.1

Simplify each term.

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Step 15.2.1.1

Cancel the common factor of $2$ .

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Step 15.2.1.1.1

Move the leading negative in $- \frac{1}{2}$ into the numerator.

$f (- \frac{1}{2}) = 4 (\frac{- 1}{2}) + \frac{1}{- \frac{1}{2}}$

Step 15.2.1.1.2

Factor $2$ out of $4$ .

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

Step 15.2.1.1.3

Cancel the common factor.

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

Step 15.2.1.1.4

Rewrite the expression.

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

Step 15.2.1.2

Multiply $2$ by $- 1$ .

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

Step 15.2.1.3

Cancel the common factor of $1$ and $- 1$ .

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Step 15.2.1.3.1

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

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

Step 15.2.1.3.2

Move the negative in front of the fraction.

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

Step 15.2.1.4

Multiply the numerator by the reciprocal of the denominator.

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

Step 15.2.1.5

Multiply $- (1 \cdot 2)$ .

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Step 15.2.1.5.1

Multiply $2$ by $1$ .

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

Step 15.2.1.5.2

Multiply $- 1$ by $2$ .

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

Step 15.2.2

Subtract $2$ from $- 2$ .

$f (- \frac{1}{2}) = - 4$

Step 15.2.3

The final answer is $- 4$ .

$y = - 4$

Step 16

These are the local extrema for $f (x) = 4 x + \frac{1}{x}$ .

$(\frac{1}{2}, 4)$ is a local minima

$(- \frac{1}{2}, - 4)$ is a local maxima

Step 17