Calculus Examples

$4 x - 64 x^{- 3}$

Step 1

Write $4 x - 64 x^{- 3}$ as a function.

$f (x) = 4 x - 64 x^{- 3}$

Step 2

Find the first derivative of the function.

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Step 2.1

By the Sum Rule, the derivative of $4 x - 64 x^{- 3}$ with respect to $x$ is $\frac{d}{d x} [4 x] + \frac{d}{d x} [- 64 x^{- 3}]$ .

$\frac{d}{d x} [4 x] + \frac{d}{d x} [- 64 x^{- 3}]$

Step 2.2

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

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Step 2.2.1

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

$4 \frac{d}{d x} [x] + \frac{d}{d x} [- 64 x^{- 3}]$

Step 2.2.2

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

$4 \cdot 1 + \frac{d}{d x} [- 64 x^{- 3}]$

Step 2.2.3

Multiply $4$ by $1$ .

$4 + \frac{d}{d x} [- 64 x^{- 3}]$

Step 2.3

Evaluate $\frac{d}{d x} [- 64 x^{- 3}]$ .

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Step 2.3.1

Since $- 64$ is constant with respect to $x$ , the derivative of $- 64 x^{- 3}$ with respect to $x$ is $- 64 \frac{d}{d x} [x^{- 3}]$ .

$4 - 64 \frac{d}{d x} [x^{- 3}]$

Step 2.3.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = - 3$ .

$4 - 64 (- 3 x^{- 4})$

Step 2.3.3

Multiply $- 3$ by $- 64$ .

$4 + 192 x^{- 4}$

Step 2.4

Simplify.

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Step 2.4.1

Reorder terms.

$192 x^{- 4} + 4$

Step 2.4.2

Simplify each term.

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Step 2.4.2.1

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

$192 \frac{1}{x^{4}} + 4$

Step 2.4.2.2

Combine $192$ and $\frac{1}{x^{4}}$ .

$\frac{192}{x^{4}} + 4$

Step 3

Find the second derivative of the function.

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Step 3.1

By the Sum Rule, the derivative of $\frac{192}{x^{4}} + 4$ with respect to $x$ is $\frac{d}{d x} [\frac{192}{x^{4}}] + \frac{d}{d x} [4]$ .

$f'' (x) = \frac{d}{d x} (\frac{192}{x^{4}}) + \frac{d}{d x} (4)$

Step 3.2

Evaluate $\frac{d}{d x} [\frac{192}{x^{4}}]$ .

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Step 3.2.1

Since $192$ is constant with respect to $x$ , the derivative of $\frac{192}{x^{4}}$ with respect to $x$ is $192 \frac{d}{d x} [\frac{1}{x^{4}}]$ .

$f'' (x) = 192 \frac{d}{d x} (\frac{1}{x^{4}}) + \frac{d}{d x} (4)$

Step 3.2.2

Rewrite $\frac{1}{x^{4}}$ as ${(x^{4})}^{- 1}$ .

$f'' (x) = 192 \frac{d}{d x} ({(x^{4})}^{- 1}) + \frac{d}{d x} (4)$

Step 3.2.3

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 $g (x) = x^{4}$ .

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Step 3.2.3.1

To apply the Chain Rule, set $u$ as $x^{4}$ .

$f'' (x) = 192 (\frac{d}{d u} (u^{- 1}) \frac{d}{d x} (x^{4})) + \frac{d}{d x} (4)$

Step 3.2.3.2

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

$f'' (x) = 192 (- u^{- 2} \frac{d}{d x} x^{4}) + \frac{d}{d x} (4)$

Step 3.2.3.3

Replace all occurrences of $u$ with $x^{4}$ .

$f'' (x) = 192 (- {(x^{4})}^{- 2} \frac{d}{d x} x^{4}) + \frac{d}{d x} (4)$

Step 3.2.4

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 4$ .

$f'' (x) = 192 (- {(x^{4})}^{- 2} (4 x^{3})) + \frac{d}{d x} (4)$

Step 3.2.5

Multiply the exponents in ${(x^{4})}^{- 2}$ .

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Step 3.2.5.1

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

$f'' (x) = 192 (- x^{4 \cdot - 2} (4 x^{3})) + \frac{d}{d x} (4)$

Step 3.2.5.2

Multiply $4$ by $- 2$ .

$f'' (x) = 192 (- x^{- 8} (4 x^{3})) + \frac{d}{d x} (4)$

Step 3.2.6

Multiply $4$ by $- 1$ .

$f'' (x) = 192 (- 4 x^{- 8} x^{3}) + \frac{d}{d x} (4)$

Step 3.2.7

Multiply $x^{- 8}$ by $x^{3}$ by adding the exponents.

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Step 3.2.7.1

Move $x^{3}$ .

$f'' (x) = 192 (- 4 (x^{3} x^{- 8})) + \frac{d}{d x} (4)$

Step 3.2.7.2

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

$f'' (x) = 192 (- 4 x^{3 - 8}) + \frac{d}{d x} (4)$

Step 3.2.7.3

Subtract $8$ from $3$ .

$f'' (x) = 192 (- 4 x^{- 5}) + \frac{d}{d x} (4)$

Step 3.2.8

Multiply $- 4$ by $192$ .

$f'' (x) = - 768 x^{- 5} + \frac{d}{d x} (4)$

Step 3.3

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

$f'' (x) = - 768 x^{- 5} + 0$

Step 3.4

Simplify.

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Step 3.4.1

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

$f'' (x) = - 768 \frac{1}{x^{5}} + 0$

Step 3.4.2

Combine terms.

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Step 3.4.2.1

Combine $- 768$ and $\frac{1}{x^{5}}$ .

$f'' (x) = \frac{- 768}{x^{5}} + 0$

Step 3.4.2.2

Move the negative in front of the fraction.

$f'' (x) = - \frac{768}{x^{5}} + 0$

Step 3.4.2.3

Add $- \frac{768}{x^{5}}$ and $0$ .

$f'' (x) = - \frac{768}{x^{5}}$

Step 4

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

$\frac{192}{x^{4}} + 4 = 0$

Step 5

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

No Local Extrema

Step 6

No Local Extrema

Step 7