Calculus Examples

$f (x) = 2 x^{2} - \frac{6}{x^{2}}$

Step 1

Find the first derivative of the function.

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Step 1.1

By the Sum Rule, the derivative of $2 x^{2} - \frac{6}{x^{2}}$ with respect to $x$ is $\frac{d}{d x} [2 x^{2}] + \frac{d}{d x} [- \frac{6}{x^{2}}]$ .

$\frac{d}{d x} [2 x^{2}] + \frac{d}{d x} [- \frac{6}{x^{2}}]$

Step 1.2

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

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Step 1.2.1

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

$2 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [- \frac{6}{x^{2}}]$

Step 1.2.2

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

$2 (2 x) + \frac{d}{d x} [- \frac{6}{x^{2}}]$

Step 1.2.3

Multiply $2$ by $2$ .

$4 x + \frac{d}{d x} [- \frac{6}{x^{2}}]$

Step 1.3

Evaluate $\frac{d}{d x} [- \frac{6}{x^{2}}]$ .

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Step 1.3.1

Since $- 6$ is constant with respect to $x$ , the derivative of $- \frac{6}{x^{2}}$ with respect to $x$ is $- 6 \frac{d}{d x} [\frac{1}{x^{2}}]$ .

$4 x - 6 \frac{d}{d x} [\frac{1}{x^{2}}]$

Step 1.3.2

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

$4 x - 6 \frac{d}{d x} [{(x^{2})}^{- 1}]$

Step 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 1.3.3.1

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

$4 x - 6 (\frac{d}{d u} [u^{- 1}] \frac{d}{d x} [x^{2}])$

Step 1.3.3.2

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

$4 x - 6 (- u^{- 2} \frac{d}{d x} [x^{2}])$

Step 1.3.3.3

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

$4 x - 6 (- {(x^{2})}^{- 2} \frac{d}{d x} [x^{2}])$

Step 1.3.4

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

$4 x - 6 (- {(x^{2})}^{- 2} (2 x))$

Step 1.3.5

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

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Step 1.3.5.1

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

$4 x - 6 (- x^{2 \cdot - 2} (2 x))$

Step 1.3.5.2

Multiply $2$ by $- 2$ .

$4 x - 6 (- x^{- 4} (2 x))$

Step 1.3.6

Multiply $2$ by $- 1$ .

$4 x - 6 (- 2 x^{- 4} x)$

Step 1.3.7

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

$4 x - 6 (- 2 (x^{1} x^{- 4}))$

Step 1.3.8

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

$4 x - 6 (- 2 x^{1 - 4})$

Step 1.3.9

Subtract $4$ from $1$ .

$4 x - 6 (- 2 x^{- 3})$

Step 1.3.10

Multiply $- 2$ by $- 6$ .

$4 x + 12 x^{- 3}$

Step 1.4

Simplify.

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Step 1.4.1

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

$4 x + 12 \frac{1}{x^{3}}$

Step 1.4.2

Combine $12$ and $\frac{1}{x^{3}}$ .

$4 x + \frac{12}{x^{3}}$

Step 2

Find the second derivative of the function.

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Step 2.1

By the Sum Rule, the derivative of $4 x + \frac{12}{x^{3}}$ with respect to $x$ is $\frac{d}{d x} [4 x] + \frac{d}{d x} [\frac{12}{x^{3}}]$ .

$f'' (x) = \frac{d}{d x} (4 x) + \frac{d}{d x} (\frac{12}{x^{3}})$

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

$f'' (x) = 4 \frac{d}{d x} (x) + \frac{d}{d x} (\frac{12}{x^{3}})$

Step 2.2.2

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

$f'' (x) = 4 \cdot 1 + \frac{d}{d x} (\frac{12}{x^{3}})$

Step 2.2.3

Multiply $4$ by $1$ .

$f'' (x) = 4 + \frac{d}{d x} (\frac{12}{x^{3}})$

Step 2.3

Evaluate $\frac{d}{d x} [\frac{12}{x^{3}}]$ .

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Step 2.3.1

Since $12$ is constant with respect to $x$ , the derivative of $\frac{12}{x^{3}}$ with respect to $x$ is $12 \frac{d}{d x} [\frac{1}{x^{3}}]$ .

$f'' (x) = 4 + 12 \frac{d}{d x} (\frac{1}{x^{3}})$

Step 2.3.2

Rewrite $\frac{1}{x^{3}}$ as ${(x^{3})}^{- 1}$ .

$f'' (x) = 4 + 12 \frac{d}{d x} ({(x^{3})}^{- 1})$

Step 2.3.3

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 $g (x) = x^{3}$ .

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Step 2.3.3.1

To apply the Chain Rule, set $u$ as $x^{3}$ .

$f'' (x) = 4 + 12 (\frac{d}{d u} (u^{- 1}) \frac{d}{d x} (x^{3}))$

Step 2.3.3.2

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

$f'' (x) = 4 + 12 (- u^{- 2} \frac{d}{d x} x^{3})$

Step 2.3.3.3

Replace all occurrences of $u$ with $x^{3}$ .

$f'' (x) = 4 + 12 (- {(x^{3})}^{- 2} \frac{d}{d x} x^{3})$

Step 2.3.4

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

$f'' (x) = 4 + 12 (- {(x^{3})}^{- 2} (3 x^{2}))$

Step 2.3.5

Multiply the exponents in ${(x^{3})}^{- 2}$ .

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Step 2.3.5.1

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

$f'' (x) = 4 + 12 (- x^{3 \cdot - 2} (3 x^{2}))$

Step 2.3.5.2

Multiply $3$ by $- 2$ .

$f'' (x) = 4 + 12 (- x^{- 6} (3 x^{2}))$

Step 2.3.6

Multiply $3$ by $- 1$ .

$f'' (x) = 4 + 12 (- 3 x^{- 6} x^{2})$

Step 2.3.7

Multiply $x^{- 6}$ by $x^{2}$ by adding the exponents.

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Step 2.3.7.1

Move $x^{2}$ .

$f'' (x) = 4 + 12 (- 3 (x^{2} x^{- 6}))$

Step 2.3.7.2

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

$f'' (x) = 4 + 12 (- 3 x^{2 - 6})$

Step 2.3.7.3

Subtract $6$ from $2$ .

$f'' (x) = 4 + 12 (- 3 x^{- 4})$

Step 2.3.8

Multiply $- 3$ by $12$ .

$f'' (x) = 4 - 36 x^{- 4}$

Step 2.4

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

$f'' (x) = 4 - 36 \frac{1}{x^{4}}$

Step 2.5

Simplify.

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Step 2.5.1

Combine terms.

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Step 2.5.1.1

Combine $- 36$ and $\frac{1}{x^{4}}$ .

$f'' (x) = 4 + \frac{- 36}{x^{4}}$

Step 2.5.1.2

Move the negative in front of the fraction.

$f'' (x) = 4 - \frac{36}{x^{4}}$

Step 2.5.2

Reorder terms.

$f'' (x) = - \frac{36}{x^{4}} + 4$

Step 3

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

$4 x + \frac{12}{x^{3}} = 0$

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