Calculus Examples

$\frac{1}{x} - \frac{2}{x^{2}}$ , $[- 2, 1]$

Step 1

Find the critical points.

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Step 1.1

Find the first derivative.

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Step 1.1.1

Find the first derivative.

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Step 1.1.1.1

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

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

Step 1.1.1.2

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

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Step 1.1.1.2.1

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

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

Step 1.1.1.2.2

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

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

Step 1.1.1.3

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

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Step 1.1.1.3.1

Since $- 2$ is constant with respect to $x$ , the derivative of $- \frac{2}{x^{2}}$ with respect to $x$ is $- 2 \frac{d}{d x} [\frac{1}{x^{2}}]$ .

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

Step 1.1.1.3.2

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

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

Step 1.1.1.3.3

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 1.1.1.3.3.1

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

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

Step 1.1.1.3.3.2

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

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

Step 1.1.1.3.3.3

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

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

Step 1.1.1.3.4

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

$- x^{- 2} - 2 (- {(x^{2})}^{- 2} (2 x))$

Step 1.1.1.3.5

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

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Step 1.1.1.3.5.1

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

$- x^{- 2} - 2 (- x^{2 \cdot - 2} (2 x))$

Step 1.1.1.3.5.2

Multiply $2$ by $- 2$ .

$- x^{- 2} - 2 (- x^{- 4} (2 x))$

Step 1.1.1.3.6

Multiply $2$ by $- 1$ .

$- x^{- 2} - 2 (- 2 x^{- 4} x)$

Step 1.1.1.3.7

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

$- x^{- 2} - 2 (- 2 (x^{1} x^{- 4}))$

Step 1.1.1.3.8

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

$- x^{- 2} - 2 (- 2 x^{1 - 4})$

Step 1.1.1.3.9

Subtract $4$ from $1$ .

$- x^{- 2} - 2 (- 2 x^{- 3})$

Step 1.1.1.3.10

Multiply $- 2$ by $- 2$ .

$- x^{- 2} + 4 x^{- 3}$

Step 1.1.1.4

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

$- \frac{1}{x^{2}} + 4 x^{- 3}$

Step 1.1.1.5

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

$- \frac{1}{x^{2}} + 4 \frac{1}{x^{3}}$

Step 1.1.1.6

Combine $4$ and $\frac{1}{x^{3}}$ .

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

Step 1.1.2

The first derivative of $f (x)$ with respect to $x$ is $- \frac{1}{x^{2}} + \frac{4}{x^{3}}$ .

$- \frac{1}{x^{2}} + \frac{4}{x^{3}}$

Step 1.2

Set the first derivative equal to $0$ then solve the equation $- \frac{1}{x^{2}} + \frac{4}{x^{3}} = 0$ .

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Step 1.2.1

Set the first derivative equal to $0$ .

$- \frac{1}{x^{2}} + \frac{4}{x^{3}} = 0$

Step 1.2.2

Find the LCD of the terms in the equation.

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Step 1.2.2.1

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

$x^{2}, x^{3}, 1$

Step 1.2.2.2

Since $x^{2}, x^{3}, 1$ contains both numbers and variables, there are two steps to find the LCM. Find LCM for the numeric part $1, 1, 1$ then find LCM for the variable part $x^{2}, x^{3}$ .

Step 1.2.2.3

The LCM is the smallest positive number that all of the numbers divide into evenly.

1. List the prime factors of each number.

2. Multiply each factor the greatest number of times it occurs in either number.

Step 1.2.2.4

The number $1$ is not a prime number because it only has one positive factor, which is itself.

Not prime

Step 1.2.2.5

The LCM of $1, 1, 1$ is the result of multiplying all prime factors the greatest number of times they occur in either number.

$1$

Step 1.2.2.6

The factors for $x^{2}$ are $x \cdot x$ , which is $x$ multiplied by each other $2$ times.

$x^{2} = x \cdot x$

$x$ occurs $2$ times.

Step 1.2.2.7

The factors for $x^{3}$ are $x \cdot x \cdot x$ , which is $x$ multiplied by each other $3$ times.

$x^{3} = x \cdot x \cdot x$

$x$ occurs $3$ times.

Step 1.2.2.8

The LCM of $x^{2}, x^{3}$ is the result of multiplying all prime factors the greatest number of times they occur in either term.

$x \cdot x \cdot x$

Step 1.2.2.9

Simplify $x \cdot x \cdot x$ .

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Step 1.2.2.9.1

Multiply $x$ by $x$ .

$x^{2} \cdot x$

Step 1.2.2.9.2

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

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Step 1.2.2.9.2.1

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

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Step 1.2.2.9.2.1.1

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

$x^{2} \cdot x^{1}$

Step 1.2.2.9.2.1.2

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

$x^{2 + 1}$

Step 1.2.2.9.2.2

Add $2$ and $1$ .

$x^{3}$

Step 1.2.3

Multiply each term in $- \frac{1}{x^{2}} + \frac{4}{x^{3}} = 0$ by $x^{3}$ to eliminate the fractions.

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Step 1.2.3.1

Multiply each term in $- \frac{1}{x^{2}} + \frac{4}{x^{3}} = 0$ by $x^{3}$ .

$- \frac{1}{x^{2}} x^{3} + \frac{4}{x^{3}} x^{3} = 0 x^{3}$

Step 1.2.3.2

Simplify the left side.

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Step 1.2.3.2.1

Simplify each term.

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Step 1.2.3.2.1.1

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

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Step 1.2.3.2.1.1.1

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

$\frac{- 1}{x^{2}} x^{3} + \frac{4}{x^{3}} x^{3} = 0 x^{3}$

Step 1.2.3.2.1.1.2

Factor $x^{2}$ out of $x^{3}$ .

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

Step 1.2.3.2.1.1.3

Cancel the common factor.

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

Step 1.2.3.2.1.1.4

Rewrite the expression.

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

Step 1.2.3.2.1.2

Cancel the common factor of $x^{3}$ .

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Step 1.2.3.2.1.2.1

Cancel the common factor.

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

Step 1.2.3.2.1.2.2

Rewrite the expression.

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

Step 1.2.3.3

Simplify the right side.

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Step 1.2.3.3.1

Multiply $0$ by $x^{3}$ .

$- x + 4 = 0$

Step 1.2.4

Solve the equation.

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Step 1.2.4.1

Subtract $4$ from both sides of the equation.

$- x = - 4$

Step 1.2.4.2

Divide each term in $- x = - 4$ by $- 1$ and simplify.

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Step 1.2.4.2.1

Divide each term in $- x = - 4$ by $- 1$ .

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

Step 1.2.4.2.2

Simplify the left side.

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Step 1.2.4.2.2.1

Dividing two negative values results in a positive value.

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

Step 1.2.4.2.2.2

Divide $x$ by $1$ .

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

Step 1.2.4.2.3

Simplify the right side.

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Step 1.2.4.2.3.1

Divide $- 4$ by $- 1$ .

$x = 4$

Step 1.3

Find the values where the derivative is undefined.

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Step 1.3.1

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

$x^{2} = 0$

Step 1.3.2

Solve for $x$ .

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Step 1.3.2.1

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

$x = \pm \sqrt{0}$

Step 1.3.2.2

Simplify $\pm \sqrt{0}$ .

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Step 1.3.2.2.1

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

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

Step 1.3.2.2.2

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

$x = \pm 0$

Step 1.3.2.2.3

Plus or minus $0$ is $0$ .

$x = 0$

Step 1.3.3

Set the denominator in $\frac{4}{x^{3}}$ equal to $0$ to find where the expression is undefined.

$x^{3} = 0$

Step 1.3.4

Solve for $x$ .

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Step 1.3.4.1

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

$x = \sqrt[3]{0}$

Step 1.3.4.2

Simplify $\sqrt[3]{0}$ .

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Step 1.3.4.2.1

Rewrite $0$ as $0^{3}$ .

$x = \sqrt[3]{0^{3}}$

Step 1.3.4.2.2

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

$x = 0$

Step 1.4

Evaluate $\frac{1}{x} - \frac{2}{x^{2}}$ at each $x$ value where the derivative is $0$ or undefined.

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Step 1.4.1

Evaluate at $x = 4$ .

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Step 1.4.1.1

Substitute $4$ for $x$ .

$\frac{1}{4} - \frac{2}{{(4)}^{2}}$

Step 1.4.1.2

Simplify.

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Step 1.4.1.2.1

Simplify each term.

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Step 1.4.1.2.1.1

Raise $4$ to the power of $2$ .

$\frac{1}{4} - \frac{2}{16}$

Step 1.4.1.2.1.2

Cancel the common factor of $2$ and $16$ .

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Step 1.4.1.2.1.2.1

Factor $2$ out of $2$ .

$\frac{1}{4} - \frac{2 (1)}{16}$

Step 1.4.1.2.1.2.2

Cancel the common factors.

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Step 1.4.1.2.1.2.2.1

Factor $2$ out of $16$ .

$\frac{1}{4} - \frac{2 \cdot 1}{2 \cdot 8}$

Step 1.4.1.2.1.2.2.2

Cancel the common factor.

$\frac{1}{4} - \frac{2 \cdot 1}{2 \cdot 8}$

Step 1.4.1.2.1.2.2.3

Rewrite the expression.

$\frac{1}{4} - \frac{1}{8}$

Step 1.4.1.2.2

To write $\frac{1}{4}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{1}{4} \cdot \frac{2}{2} - \frac{1}{8}$

Step 1.4.1.2.3

Write each expression with a common denominator of $8$ , by multiplying each by an appropriate factor of $1$ .

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Step 1.4.1.2.3.1

Multiply $\frac{1}{4}$ by $\frac{2}{2}$ .

$\frac{2}{4 \cdot 2} - \frac{1}{8}$

Step 1.4.1.2.3.2

Multiply $4$ by $2$ .

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

Step 1.4.1.2.4

Combine the numerators over the common denominator.

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

Step 1.4.1.2.5

Subtract $1$ from $2$ .

$\frac{1}{8}$

Step 1.4.2

Evaluate at $x = 0$ .

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Step 1.4.2.1

Substitute $0$ for $x$ .

$\frac{1}{0} - \frac{2}{{(0)}^{2}}$

Step 1.4.2.2

Simplify.

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Step 1.4.2.2.1

Raising $0$ to any positive power yields $0$ .

$\frac{1}{0} - \frac{2}{0}$

Step 1.4.2.2.2

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

Undefined

Step 1.4.3

List all of the points.

$(4, \frac{1}{8})$

Step 2

Exclude the points that are not on the interval.

Step 3

Evaluate at the included endpoints.

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Step 3.1

Evaluate at $x = - 2$ .

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Step 3.1.1

Substitute $- 2$ for $x$ .

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

Step 3.1.2

Simplify.

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Step 3.1.2.1

Simplify each term.

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Step 3.1.2.1.1

Move the negative in front of the fraction.

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

Step 3.1.2.1.2

Cancel the common factor of $2$ and ${(- 2)}^{2}$ .

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Step 3.1.2.1.2.1

Rewrite $2$ as $- 1 (- 2)$ .

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

Step 3.1.2.1.2.2

Factor $- 2$ out of $- 1 (- 2)$ .

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

Step 3.1.2.1.2.3

Cancel the common factors.

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Step 3.1.2.1.2.3.1

Factor $- 2$ out of ${(- 2)}^{2}$ .

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

Step 3.1.2.1.2.3.2

Cancel the common factor.

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

Step 3.1.2.1.2.3.3

Rewrite the expression.

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

Step 3.1.2.1.3

Dividing two negative values results in a positive value.

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

Step 3.1.2.2

Combine fractions.

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Step 3.1.2.2.1

Combine the numerators over the common denominator.

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

Step 3.1.2.2.2

Simplify the expression.

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Step 3.1.2.2.2.1

Subtract $1$ from $- 1$ .

$\frac{- 2}{2}$

Step 3.1.2.2.2.2

Divide $- 2$ by $2$ .

$- 1$

Step 3.2

Evaluate at $x = 1$ .

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Step 3.2.1

Substitute $1$ for $x$ .

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

Step 3.2.2

Simplify.

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Step 3.2.2.1

Simplify each term.

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Step 3.2.2.1.1

Divide $1$ by $1$ .

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

Step 3.2.2.1.2

One to any power is one.

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

Step 3.2.2.1.3

Divide $2$ by $1$ .

$1 - 1 \cdot 2$

Step 3.2.2.1.4

Multiply $- 1$ by $2$ .

$1 - 2$

Step 3.2.2.2

Subtract $2$ from $1$ .

$- 1$

Step 3.3

List all of the points.

$(- 2, - 1), (1, - 1)$

Step 4

Compare the $f (x)$ values found for each value of $x$ in order to determine the absolute maximum and minimum over the given interval. The maximum will occur at the highest $f (x)$ value and the minimum will occur at the lowest $f (x)$ value.

Absolute Maximum: $(- 2, - 1), (1, - 1)$

Absolute Minimum: $(- 2, - 1), (1, - 1)$

Step 5