Calculus Examples

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

Step 1

Find the first derivative of the function.

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Step 1.1

Apply basic rules of exponents.

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Step 1.1.1

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

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

Step 1.1.2

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

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Step 1.1.2.1

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

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

Step 1.1.2.2

Multiply $2$ by $- 1$ .

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

Step 1.2

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

$- 2 x^{- 3}$

Step 1.3

Simplify.

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Step 1.3.1

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

$- 2 \frac{1}{x^{3}}$

Step 1.3.2

Combine terms.

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Step 1.3.2.1

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

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

Step 1.3.2.2

Move the negative in front of the fraction.

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

Step 2

Find the second derivative of the function.

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Step 2.1

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 2.2

Apply basic rules of exponents.

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Step 2.2.1

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

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

Step 2.2.2

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

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Step 2.2.2.1

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

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

Step 2.2.2.2

Multiply $3$ by $- 1$ .

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

Step 2.3

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 2.4

Multiply $- 3$ by $- 2$ .

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

Step 2.5

Simplify.

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Step 2.5.1

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

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

Step 2.5.2

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

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

Step 3

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 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