Calculus Examples

$\frac{3}{x}$

Step 1

Find the first derivative.

Tap for more steps...

Step 1.1

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

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

Step 1.2

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

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

Step 1.3

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

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

Step 1.4

Multiply $- 1$ by $3$ .

$- 3 x^{- 2}$

Step 1.5

Simplify.

Tap for more steps...

Step 1.5.1

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

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

Step 1.5.2

Combine terms.

Tap for more steps...

Step 1.5.2.1

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

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

Step 1.5.2.2

Move the negative in front of the fraction.

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

Step 2

Find the second derivative.

Tap for more steps...

Step 2.1

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

$- 3 \frac{d}{d x} [\frac{1}{x^{2}}]$

Step 2.2

Apply basic rules of exponents.

Tap for more steps...

Step 2.2.1

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

$- 3 \frac{d}{d x} [{(x^{2})}^{- 1}]$

Step 2.2.2

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

Tap for more steps...

Step 2.2.2.1

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

$- 3 \frac{d}{d x} [x^{2 \cdot - 1}]$

Step 2.2.2.2

Multiply $2$ by $- 1$ .

$- 3 \frac{d}{d x} [x^{- 2}]$

Step 2.3

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

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

Step 2.4

Multiply $- 2$ by $- 3$ .

$6 x^{- 3}$

Step 2.5

Simplify.

Tap for more steps...

Step 2.5.1

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

$6 \frac{1}{x^{3}}$

Step 2.5.2

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

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

Step 3

Find the third derivative.

Tap for more steps...

Step 3.1

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

$6 \frac{d}{d x} [\frac{1}{x^{3}}]$

Step 3.2

Apply basic rules of exponents.

Tap for more steps...

Step 3.2.1

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

$6 \frac{d}{d x} [{(x^{3})}^{- 1}]$

Step 3.2.2

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

Tap for more steps...

Step 3.2.2.1

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

$6 \frac{d}{d x} [x^{3 \cdot - 1}]$

Step 3.2.2.2

Multiply $3$ by $- 1$ .

$6 \frac{d}{d x} [x^{- 3}]$

Step 3.3

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

$6 (- 3 x^{- 4})$

Step 3.4

Multiply $- 3$ by $6$ .

$- 18 x^{- 4}$

Step 3.5

Simplify.

Tap for more steps...

Step 3.5.1

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

$- 18 \frac{1}{x^{4}}$

Step 3.5.2

Combine terms.

Tap for more steps...

Step 3.5.2.1

Combine $- 18$ and $\frac{1}{x^{4}}$ .

$\frac{- 18}{x^{4}}$

Step 3.5.2.2

Move the negative in front of the fraction.

$f''' (x) = - \frac{18}{x^{4}}$

Step 4

Find the fourth derivative.

Tap for more steps...

Step 4.1

Since $- 18$ is constant with respect to $x$ , the derivative of $- \frac{18}{x^{4}}$ with respect to $x$ is $- 18 \frac{d}{d x} [\frac{1}{x^{4}}]$ .

$- 18 \frac{d}{d x} [\frac{1}{x^{4}}]$

Step 4.2

Apply basic rules of exponents.

Tap for more steps...

Step 4.2.1

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

$- 18 \frac{d}{d x} [{(x^{4})}^{- 1}]$

Step 4.2.2

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

Tap for more steps...

Step 4.2.2.1

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

$- 18 \frac{d}{d x} [x^{4 \cdot - 1}]$

Step 4.2.2.2

Multiply $4$ by $- 1$ .

$- 18 \frac{d}{d x} [x^{- 4}]$

Step 4.3

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

$- 18 (- 4 x^{- 5})$

Step 4.4

Multiply $- 4$ by $- 18$ .

$72 x^{- 5}$

Step 4.5

Simplify.

Tap for more steps...

Step 4.5.1

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

$72 \frac{1}{x^{5}}$

Step 4.5.2

Combine $72$ and $\frac{1}{x^{5}}$ .

$f^{4} (x) = \frac{72}{x^{5}}$