Calculus Examples

$\frac{1}{x}$

Step 1

Find the first derivative of the function.

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Step 1.1

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

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

Step 1.2

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

$- x^{- 2}$

Step 1.3

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

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

Step 2

Find the second derivative of the function.

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Step 2.1

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = - 1$ and $g (x) = \frac{1}{x^{2}}$ .

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

Step 2.2

Differentiate.

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Step 2.2.1

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

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

Step 2.2.2

Multiply the exponents in ${(x^{2})}^{- 1}$ .

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Step 2.2.2.1

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

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

Step 2.2.2.2

Multiply $2$ by $- 1$ .

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

Step 2.2.3

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

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

Step 2.2.4

Multiply $- 2$ by $- 1$ .

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

Step 2.2.5

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

$f'' (x) = 2 x^{- 3} + \frac{1}{x^{2}} \cdot 0$

Step 2.2.6

Simplify the expression.

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Step 2.2.6.1

Multiply $\frac{1}{x^{2}}$ by $0$ .

$f'' (x) = 2 x^{- 3} + 0$

Step 2.2.6.2

Add $2 x^{- 3}$ and $0$ .

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

Step 2.3

Simplify.

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Step 2.3.1

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

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

Step 2.3.2

Combine $2$ and $\frac{1}{x^{3}}$ .

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

Step 3

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

$- \frac{1}{x^{2}} = 0$

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