Calculus Examples

$f (x) = 2 x - \frac{32}{x}$

Step 1

Find the first derivative of the function.

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Step 1.1

By the Sum Rule, the derivative of $2 x - \frac{32}{x}$ with respect to $x$ is $\frac{d}{d x} [2 x] + \frac{d}{d x} [- \frac{32}{x}]$ .

$\frac{d}{d x} [2 x] + \frac{d}{d x} [- \frac{32}{x}]$

Step 1.2

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

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Step 1.2.1

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

$2 \frac{d}{d x} [x] + \frac{d}{d x} [- \frac{32}{x}]$

Step 1.2.2

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

$2 \cdot 1 + \frac{d}{d x} [- \frac{32}{x}]$

Step 1.2.3

Multiply $2$ by $1$ .

$2 + \frac{d}{d x} [- \frac{32}{x}]$

Step 1.3

Evaluate $\frac{d}{d x} [- \frac{32}{x}]$ .

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Step 1.3.1

Since $- 32$ is constant with respect to $x$ , the derivative of $- \frac{32}{x}$ with respect to $x$ is $- 32 \frac{d}{d x} [\frac{1}{x}]$ .

$2 - 32 \frac{d}{d x} [\frac{1}{x}]$

Step 1.3.2

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

$2 - 32 \frac{d}{d x} [x^{- 1}]$

Step 1.3.3

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

$2 - 32 (- x^{- 2})$

Step 1.3.4

Multiply $- 1$ by $- 32$ .

$2 + 32 x^{- 2}$

Step 1.4

Simplify.

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Step 1.4.1

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

$2 + 32 \frac{1}{x^{2}}$

Step 1.4.2

Combine $32$ and $\frac{1}{x^{2}}$ .

$2 + \frac{32}{x^{2}}$

Step 1.4.3

Reorder terms.

$\frac{32}{x^{2}} + 2$

Step 2

Find the second derivative of the function.

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Step 2.1

By the Sum Rule, the derivative of $\frac{32}{x^{2}} + 2$ with respect to $x$ is $\frac{d}{d x} [\frac{32}{x^{2}}] + \frac{d}{d x} [2]$ .

$f'' (x) = \frac{d}{d x} (\frac{32}{x^{2}}) + \frac{d}{d x} (2)$

Step 2.2

Evaluate $\frac{d}{d x} [\frac{32}{x^{2}}]$ .

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Step 2.2.1

Since $32$ is constant with respect to $x$ , the derivative of $\frac{32}{x^{2}}$ with respect to $x$ is $32 \frac{d}{d x} [\frac{1}{x^{2}}]$ .

$f'' (x) = 32 \frac{d}{d x} (\frac{1}{x^{2}}) + \frac{d}{d x} (2)$

Step 2.2.2

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

$f'' (x) = 32 \frac{d}{d x} ({(x^{2})}^{- 1}) + \frac{d}{d x} (2)$

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 $x^{2}$ .

$f'' (x) = 32 (\frac{d}{d u} (u^{- 1}) \frac{d}{d x} (x^{2})) + \frac{d}{d x} (2)$

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 $n = - 1$ .

$f'' (x) = 32 (- u^{- 2} \frac{d}{d x} x^{2}) + \frac{d}{d x} (2)$

Step 2.2.3.3

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

$f'' (x) = 32 (- {(x^{2})}^{- 2} \frac{d}{d x} x^{2}) + \frac{d}{d x} (2)$

Step 2.2.4

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

$f'' (x) = 32 (- {(x^{2})}^{- 2} (2 x)) + \frac{d}{d x} (2)$

Step 2.2.5

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

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Step 2.2.5.1

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

$f'' (x) = 32 (- x^{2 \cdot - 2} (2 x)) + \frac{d}{d x} (2)$

Step 2.2.5.2

Multiply $2$ by $- 2$ .

$f'' (x) = 32 (- x^{- 4} (2 x)) + \frac{d}{d x} (2)$

Step 2.2.6

Multiply $2$ by $- 1$ .

$f'' (x) = 32 (- 2 x^{- 4} x) + \frac{d}{d x} (2)$

Step 2.2.7

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

$f'' (x) = 32 (- 2 (x \cdot x^{- 4})) + \frac{d}{d x} (2)$

Step 2.2.8

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

$f'' (x) = 32 (- 2 x^{1 - 4}) + \frac{d}{d x} (2)$

Step 2.2.9

Subtract $4$ from $1$ .

$f'' (x) = 32 (- 2 x^{- 3}) + \frac{d}{d x} (2)$

Step 2.2.10

Multiply $- 2$ by $32$ .

$f'' (x) = - 64 x^{- 3} + \frac{d}{d x} (2)$

Step 2.3

Since $2$ is constant with respect to $x$ , the derivative of $2$ with respect to $x$ is $0$ .

$f'' (x) = - 64 x^{- 3} + 0$

Step 2.4

Simplify.

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Step 2.4.1

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

$f'' (x) = - 64 \frac{1}{x^{3}} + 0$

Step 2.4.2

Combine terms.

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Step 2.4.2.1

Combine $- 64$ and $\frac{1}{x^{3}}$ .

$f'' (x) = \frac{- 64}{x^{3}} + 0$

Step 2.4.2.2

Move the negative in front of the fraction.

$f'' (x) = - \frac{64}{x^{3}} + 0$

Step 2.4.2.3

Add $- \frac{64}{x^{3}}$ and $0$ .

$f'' (x) = - \frac{64}{x^{3}}$

Step 3

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

$\frac{32}{x^{2}} + 2 = 0$

Step 4

Since there is no value of $x$ that makes the first derivative equal to $0$ , there are no local extrema.

No Local Extrema

Step 5

No Local Extrema

Step 6