Calculus Examples

$x - \frac{54}{x}$

Step 1

Write $x - \frac{54}{x}$ as a function.

$f (x) = x - \frac{54}{x}$

Step 2

Find the first derivative of the function.

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Step 2.1

Differentiate.

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Step 2.1.1

By the Sum Rule, the derivative of $x - \frac{54}{x}$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [- \frac{54}{x}]$ .

$\frac{d}{d x} [x] + \frac{d}{d x} [- \frac{54}{x}]$

Step 2.1.2

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

$1 + \frac{d}{d x} [- \frac{54}{x}]$

Step 2.2

Evaluate $\frac{d}{d x} [- \frac{54}{x}]$ .

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Step 2.2.1

Since $- 54$ is constant with respect to $x$ , the derivative of $- \frac{54}{x}$ with respect to $x$ is $- 54 \frac{d}{d x} [\frac{1}{x}]$ .

$1 - 54 \frac{d}{d x} [\frac{1}{x}]$

Step 2.2.2

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

$1 - 54 \frac{d}{d x} [x^{- 1}]$

Step 2.2.3

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

$1 - 54 (- x^{- 2})$

Step 2.2.4

Multiply $- 1$ by $- 54$ .

$1 + 54 x^{- 2}$

Step 2.3

Simplify.

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Step 2.3.1

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

$1 + 54 \frac{1}{x^{2}}$

Step 2.3.2

Combine $54$ and $\frac{1}{x^{2}}$ .

$1 + \frac{54}{x^{2}}$

Step 2.3.3

Reorder terms.

$\frac{54}{x^{2}} + 1$

Step 3

Find the second derivative of the function.

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Step 3.1

By the Sum Rule, the derivative of $\frac{54}{x^{2}} + 1$ with respect to $x$ is $\frac{d}{d x} [\frac{54}{x^{2}}] + \frac{d}{d x} [1]$ .

$f'' (x) = \frac{d}{d x} (\frac{54}{x^{2}}) + \frac{d}{d x} (1)$

Step 3.2

Evaluate $\frac{d}{d x} [\frac{54}{x^{2}}]$ .

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Step 3.2.1

Since $54$ is constant with respect to $x$ , the derivative of $\frac{54}{x^{2}}$ with respect to $x$ is $54 \frac{d}{d x} [\frac{1}{x^{2}}]$ .

$f'' (x) = 54 \frac{d}{d x} (\frac{1}{x^{2}}) + \frac{d}{d x} (1)$

Step 3.2.2

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

$f'' (x) = 54 \frac{d}{d x} ({(x^{2})}^{- 1}) + \frac{d}{d x} (1)$

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 $g (x) = x^{2}$ .

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Step 3.2.3.1

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

$f'' (x) = 54 (\frac{d}{d u} (u^{- 1}) \frac{d}{d x} (x^{2})) + \frac{d}{d x} (1)$

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 $n = - 1$ .

$f'' (x) = 54 (- u^{- 2} \frac{d}{d x} x^{2}) + \frac{d}{d x} (1)$

Step 3.2.3.3

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

$f'' (x) = 54 (- {(x^{2})}^{- 2} \frac{d}{d x} x^{2}) + \frac{d}{d x} (1)$

Step 3.2.4

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

$f'' (x) = 54 (- {(x^{2})}^{- 2} (2 x)) + \frac{d}{d x} (1)$

Step 3.2.5

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

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Step 3.2.5.1

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

$f'' (x) = 54 (- x^{2 \cdot - 2} (2 x)) + \frac{d}{d x} (1)$

Step 3.2.5.2

Multiply $2$ by $- 2$ .

$f'' (x) = 54 (- x^{- 4} (2 x)) + \frac{d}{d x} (1)$

Step 3.2.6

Multiply $2$ by $- 1$ .

$f'' (x) = 54 (- 2 x^{- 4} x) + \frac{d}{d x} (1)$

Step 3.2.7

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

$f'' (x) = 54 (- 2 (x \cdot x^{- 4})) + \frac{d}{d x} (1)$

Step 3.2.8

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

$f'' (x) = 54 (- 2 x^{1 - 4}) + \frac{d}{d x} (1)$

Step 3.2.9

Subtract $4$ from $1$ .

$f'' (x) = 54 (- 2 x^{- 3}) + \frac{d}{d x} (1)$

Step 3.2.10

Multiply $- 2$ by $54$ .

$f'' (x) = - 108 x^{- 3} + \frac{d}{d x} (1)$

Step 3.3

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

$f'' (x) = - 108 x^{- 3} + 0$

Step 3.4

Simplify.

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Step 3.4.1

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

$f'' (x) = - 108 \frac{1}{x^{3}} + 0$

Step 3.4.2

Combine terms.

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Step 3.4.2.1

Combine $- 108$ and $\frac{1}{x^{3}}$ .

$f'' (x) = \frac{- 108}{x^{3}} + 0$

Step 3.4.2.2

Move the negative in front of the fraction.

$f'' (x) = - \frac{108}{x^{3}} + 0$

Step 3.4.2.3

Add $- \frac{108}{x^{3}}$ and $0$ .

$f'' (x) = - \frac{108}{x^{3}}$

Step 4

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

$\frac{54}{x^{2}} + 1 = 0$

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