Calculus Examples

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

Step 1

Find the first derivative.

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Step 1.1

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

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

Step 1.2

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

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

Step 1.3

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

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

Step 1.4

Multiply $- 1$ by $2$ .

$- 2 x^{- 2}$

Step 1.5

Simplify.

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Step 1.5.1

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

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

Step 1.5.2

Combine terms.

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Step 1.5.2.1

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

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

Step 1.5.2.2

Move the negative in front of the fraction.

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

Step 2

Find the second derivative.

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Step 2.1

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

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

Step 2.2

Apply basic rules of exponents.

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Step 2.2.1

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

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

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

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

Step 2.2.2.2

Multiply $2$ by $- 1$ .

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

Step 2.3

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

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

Step 2.4

Multiply $- 2$ by $- 2$ .

$4 x^{- 3}$

Step 2.5

Simplify.

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Step 2.5.1

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

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

Step 2.5.2

Combine $4$ and $\frac{1}{x^{3}}$ .

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

Step 3

Find the third derivative.

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Step 3.1

Since $4$ is constant with respect to $x$ , the derivative of $\frac{4}{x^{3}}$ with respect to $x$ is $4 \frac{d}{d x} [\frac{1}{x^{3}}]$ .

$4 \frac{d}{d x} [\frac{1}{x^{3}}]$

Step 3.2

Apply basic rules of exponents.

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Step 3.2.1

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

$4 \frac{d}{d x} [{(x^{3})}^{- 1}]$

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

$4 \frac{d}{d x} [x^{3 \cdot - 1}]$

Step 3.2.2.2

Multiply $3$ by $- 1$ .

$4 \frac{d}{d x} [x^{- 3}]$

Step 3.3

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

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

Step 3.4

Multiply $- 3$ by $4$ .

$- 12 x^{- 4}$

Step 3.5

Simplify.

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Step 3.5.1

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

$- 12 \frac{1}{x^{4}}$

Step 3.5.2

Combine terms.

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Step 3.5.2.1

Combine $- 12$ and $\frac{1}{x^{4}}$ .

$\frac{- 12}{x^{4}}$

Step 3.5.2.2

Move the negative in front of the fraction.

$f''' (x) = - \frac{12}{x^{4}}$

Step 4

Find the fourth derivative.

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Step 4.1

Since $- 12$ is constant with respect to $x$ , the derivative of $- \frac{12}{x^{4}}$ with respect to $x$ is $- 12 \frac{d}{d x} [\frac{1}{x^{4}}]$ .

$- 12 \frac{d}{d x} [\frac{1}{x^{4}}]$

Step 4.2

Apply basic rules of exponents.

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Step 4.2.1

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

$- 12 \frac{d}{d x} [{(x^{4})}^{- 1}]$

Step 4.2.2

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

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Step 4.2.2.1

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

$- 12 \frac{d}{d x} [x^{4 \cdot - 1}]$

Step 4.2.2.2

Multiply $4$ by $- 1$ .

$- 12 \frac{d}{d x} [x^{- 4}]$

Step 4.3

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

$- 12 (- 4 x^{- 5})$

Step 4.4

Multiply $- 4$ by $- 12$ .

$48 x^{- 5}$

Step 4.5

Simplify.

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Step 4.5.1

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

$48 \frac{1}{x^{5}}$

Step 4.5.2

Combine $48$ and $\frac{1}{x^{5}}$ .

$f^{4} (x) = \frac{48}{x^{5}}$

Step 5

The fourth derivative of $f (x)$ with respect to $x$ is $\frac{48}{x^{5}}$ .

$\frac{48}{x^{5}}$