Calculus Examples

$f (x) = \frac{1}{12 x^{2}}$

Step 1

Find the first derivative.

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Step 1.1

Since $\frac{1}{12}$ is constant with respect to $x$ , the derivative of $\frac{1}{12 x^{2}}$ with respect to $x$ is $\frac{1}{12} \frac{d}{d x} [\frac{1}{x^{2}}]$ .

$f' (x) = \frac{1}{12} \cdot \frac{d}{d x} (\frac{1}{x^{2}})$

Step 1.2

Apply basic rules of exponents.

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Step 1.2.1

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

$f' (x) = \frac{1}{12} \cdot \frac{d}{d x} ({(x^{2})}^{- 1})$

Step 1.2.2

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

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Step 1.2.2.1

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

$f' (x) = \frac{1}{12} \cdot \frac{d}{d x} (x^{2 \cdot - 1})$

Step 1.2.2.2

Multiply $2$ by $- 1$ .

$f' (x) = \frac{1}{12} \cdot \frac{d}{d x} (x^{- 2})$

Step 1.3

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

$f' (x) = \frac{1}{12} \cdot (- 2 x^{- 3})$

Step 1.4

Simplify terms.

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Step 1.4.1

Combine $- 2$ and $\frac{1}{12}$ .

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

Step 1.4.2

Combine $\frac{- 2}{12}$ and $x^{- 3}$ .

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

Step 1.4.3

Move $x^{- 3}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 1.4.4

Cancel the common factor of $- 2$ and $12$ .

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Step 1.4.4.1

Factor $2$ out of $- 2$ .

$f' (x) = \frac{2 (- 1)}{12 x^{3}}$

Step 1.4.4.2

Cancel the common factors.

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Step 1.4.4.2.1

Factor $2$ out of $12 x^{3}$ .

$f' (x) = \frac{2 (- 1)}{2 (6 x^{3})}$

Step 1.4.4.2.2

Cancel the common factor.

$f' (x) = \frac{2 \cdot - 1}{2 (6 x^{3})}$

Step 1.4.4.2.3

Rewrite the expression.

$f' (x) = \frac{- 1}{6 x^{3}}$

Step 1.4.5

Move the negative in front of the fraction.

$f' (x) = - \frac{1}{6 x^{3}}$

Step 2

Find the second derivative.

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Step 2.1

Since $- \frac{1}{6}$ is constant with respect to $x$ , the derivative of $- \frac{1}{6 x^{3}}$ with respect to $x$ is $- \frac{1}{6} \frac{d}{d x} [\frac{1}{x^{3}}]$ .

$f'' (x) = - \frac{1}{6} \cdot \frac{d}{d x} \frac{1}{x^{3}}$

Step 2.2

Apply basic rules of exponents.

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Step 2.2.1

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

$f'' (x) = - \frac{1}{6} \cdot \frac{d}{d x} {(x^{3})}^{- 1}$

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, ${(a^{m})}^{n} = a^{m n}$ .

$f'' (x) = - \frac{1}{6} \cdot \frac{d}{d x} x^{3 \cdot - 1}$

Step 2.2.2.2

Multiply $3$ by $- 1$ .

$f'' (x) = - \frac{1}{6} \cdot \frac{d}{d x} x^{- 3}$

Step 2.3

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

$f'' (x) = - \frac{1}{6} \cdot (- 3 x^{- 4})$

Step 2.4

Simplify terms.

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Step 2.4.1

Multiply $- 3$ by $- 1$ .

$f'' (x) = 3 (\frac{1}{6}) \cdot x^{- 4}$

Step 2.4.2

Combine $3$ and $\frac{1}{6}$ .

$f'' (x) = \frac{3}{6} x^{- 4}$

Step 2.4.3

Combine $\frac{3}{6}$ and $x^{- 4}$ .

$f'' (x) = \frac{3 x^{- 4}}{6}$

Step 2.4.4

Move $x^{- 4}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$f'' (x) = \frac{3}{6 x^{4}}$

Step 2.4.5

Cancel the common factor of $3$ and $6$ .

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Step 2.4.5.1

Factor $3$ out of $3$ .

$f'' (x) = \frac{3 (1)}{6 x^{4}}$

Step 2.4.5.2

Cancel the common factors.

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Step 2.4.5.2.1

Factor $3$ out of $6 x^{4}$ .

$f'' (x) = \frac{3 (1)}{3 (2 x^{4})}$

Step 2.4.5.2.2

Cancel the common factor.

$f'' (x) = \frac{3 \cdot 1}{3 (2 x^{4})}$

Step 2.4.5.2.3

Rewrite the expression.

$f'' (x) = \frac{1}{2 x^{4}}$

Step 3

Find the third derivative.

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Step 3.1

Since $\frac{1}{2}$ is constant with respect to $x$ , the derivative of $\frac{1}{2 x^{4}}$ with respect to $x$ is $\frac{1}{2} \frac{d}{d x} [\frac{1}{x^{4}}]$ .

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

Step 3.2

Apply basic rules of exponents.

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Step 3.2.1

Rewrite $\frac{1}{x^{4}}$ as ${(x^{4})}^{- 1}$ .

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

Step 3.2.2

Multiply the exponents in ${(x^{4})}^{- 1}$ .

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Step 3.2.2.1

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

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

Step 3.2.2.2

Multiply $4$ by $- 1$ .

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

Step 3.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = - 4$ .

$f''' (x) = \frac{1}{2} \cdot (- 4 x^{- 5})$

Step 3.4

Simplify terms.

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Step 3.4.1

Combine $- 4$ and $\frac{1}{2}$ .

$f''' (x) = \frac{- 4}{2} x^{- 5}$

Step 3.4.2

Combine $\frac{- 4}{2}$ and $x^{- 5}$ .

$f''' (x) = \frac{- 4 x^{- 5}}{2}$

Step 3.4.3

Move $x^{- 5}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$f''' (x) = \frac{- 4}{2 x^{5}}$

Step 3.4.4

Cancel the common factor of $- 4$ and $2$ .

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Step 3.4.4.1

Factor $2$ out of $- 4$ .

$f''' (x) = \frac{2 \cdot - 2}{2 x^{5}}$

Step 3.4.4.2

Cancel the common factors.

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Step 3.4.4.2.1

Factor $2$ out of $2 x^{5}$ .

$f''' (x) = \frac{2 \cdot - 2}{2 (x^{5})}$

Step 3.4.4.2.2

Cancel the common factor.

$f''' (x) = \frac{2 \cdot - 2}{2 x^{5}}$

Step 3.4.4.2.3

Rewrite the expression.

$f''' (x) = \frac{- 2}{x^{5}}$

Step 3.4.5

Move the negative in front of the fraction.

$f''' (x) = - \frac{2}{x^{5}}$

Step 4

Find the fourth derivative.

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Step 4.1

Since $- 2$ is constant with respect to $x$ , the derivative of $- \frac{2}{x^{5}}$ with respect to $x$ is $- 2 \frac{d}{d x} [\frac{1}{x^{5}}]$ .

$f^{4} (x) = - 2 \frac{d}{d x} \frac{1}{x^{5}}$

Step 4.2

Apply basic rules of exponents.

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Step 4.2.1

Rewrite $\frac{1}{x^{5}}$ as ${(x^{5})}^{- 1}$ .

$f^{4} (x) = - 2 \frac{d}{d x} {(x^{5})}^{- 1}$

Step 4.2.2

Multiply the exponents in ${(x^{5})}^{- 1}$ .

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Step 4.2.2.1

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

$f^{4} (x) = - 2 \frac{d}{d x} x^{5 \cdot - 1}$

Step 4.2.2.2

Multiply $5$ by $- 1$ .

$f^{4} (x) = - 2 \frac{d}{d x} x^{- 5}$

Step 4.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = - 5$ .

$f^{4} (x) = - 2 (- 5 x^{- 6})$

Step 4.4

Multiply $- 5$ by $- 2$ .

$f^{4} (x) = 10 x^{- 6}$

Step 4.5

Simplify.

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Step 4.5.1

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

$f^{4} (x) = 10 (\frac{1}{x^{6}})$

Step 4.5.2

Combine $10$ and $\frac{1}{x^{6}}$ .

$f^{4} (x) = \frac{10}{x^{6}}$

Step 5

The fourth derivative of $f (x)$ with respect to $x$ is $\frac{10}{x^{6}}$ .

$\frac{10}{x^{6}}$