Calculus Examples

$y = {(\frac{1}{x})}^{2}$

Step 1

Write $y = {(\frac{1}{x})}^{2}$ as a function.

$f (x) = {(\frac{1}{x})}^{2}$

Step 2

Find the first derivative of the function.

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Step 2.1

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{2}$ and $g (x) = \frac{1}{x}$ .

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Step 2.1.1

To apply the Chain Rule, set $u$ as $\frac{1}{x}$ .

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

Step 2.1.2

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

$2 u \frac{d}{d x} [\frac{1}{x}]$

Step 2.1.3

Replace all occurrences of $u$ with $\frac{1}{x}$ .

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

Step 2.2

Differentiate using the Power Rule.

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Step 2.2.1

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

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

Step 2.2.2

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

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

Step 2.2.3

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

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

Step 2.2.4

Combine fractions.

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Step 2.2.4.1

Combine $\frac{2}{x}$ and $x^{- 2}$ .

$- \frac{2 x^{- 2}}{x}$

Step 2.2.4.2

Move $x^{- 2}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$- \frac{2}{x \cdot x^{2}}$

Step 2.3

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

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Step 2.3.1

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

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Step 2.3.1.1

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

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

Step 2.3.1.2

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

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

Step 2.3.2

Add $1$ and $2$ .

$- \frac{2}{x^{3}}$

Step 3

Find the second derivative of the function.

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Step 3.1

Since $- 2$ is constant with respect to $x$ , the derivative of $- \frac{2}{x^{3}}$ with respect to $x$ is $- 2 \frac{d}{d x} [\frac{1}{x^{3}}]$ .

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

Step 3.2

Apply basic rules of exponents.

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Step 3.2.1

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

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

Step 3.2.2

Multiply the exponents in ${(x^{3})}^{- 1}$ .

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Step 3.2.2.1

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

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

Step 3.2.2.2

Multiply $3$ by $- 1$ .

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

Step 3.3

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

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

Step 3.4

Multiply $- 3$ by $- 2$ .

$f'' (x) = 6 x^{- 4}$

Step 3.5

Simplify.

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Step 3.5.1

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

$f'' (x) = 6 (\frac{1}{x^{4}})$

Step 3.5.2

Combine $6$ and $\frac{1}{x^{4}}$ .

$f'' (x) = \frac{6}{x^{4}}$

Step 4

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

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

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