Calculus Examples

$2 x^{\frac{1}{5}} + 3$

Step 1

Find the first derivative.

Tap for more steps...

Step 1.1

By the Sum Rule, the derivative of $2 x^{\frac{1}{5}} + 3$ with respect to $x$ is $\frac{d}{d x} [2 x^{\frac{1}{5}}] + \frac{d}{d x} [3]$ .

$\frac{d}{d x} [2 x^{\frac{1}{5}}] + \frac{d}{d x} [3]$

Step 1.2

Evaluate $\frac{d}{d x} [2 x^{\frac{1}{5}}]$ .

Tap for more steps...

Step 1.2.1

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

$2 \frac{d}{d x} [x^{\frac{1}{5}}] + \frac{d}{d x} [3]$

Step 1.2.2

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

$2 (\frac{1}{5} x^{\frac{1}{5} - 1}) + \frac{d}{d x} [3]$

Step 1.2.3

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$2 (\frac{1}{5} x^{\frac{1}{5} - 1 \cdot \frac{5}{5}}) + \frac{d}{d x} [3]$

Step 1.2.4

Combine $- 1$ and $\frac{5}{5}$ .

$2 (\frac{1}{5} x^{\frac{1}{5} + \frac{- 1 \cdot 5}{5}}) + \frac{d}{d x} [3]$

Step 1.2.5

Combine the numerators over the common denominator.

$2 (\frac{1}{5} x^{\frac{1 - 1 \cdot 5}{5}}) + \frac{d}{d x} [3]$

Step 1.2.6

Simplify the numerator.

Tap for more steps...

Step 1.2.6.1

Multiply $- 1$ by $5$ .

$2 (\frac{1}{5} x^{\frac{1 - 5}{5}}) + \frac{d}{d x} [3]$

Step 1.2.6.2

Subtract $5$ from $1$ .

$2 (\frac{1}{5} x^{\frac{- 4}{5}}) + \frac{d}{d x} [3]$

Step 1.2.7

Move the negative in front of the fraction.

$2 (\frac{1}{5} x^{- \frac{4}{5}}) + \frac{d}{d x} [3]$

Step 1.2.8

Combine $\frac{1}{5}$ and $x^{- \frac{4}{5}}$ .

$2 \frac{x^{- \frac{4}{5}}}{5} + \frac{d}{d x} [3]$

Step 1.2.9

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

$\frac{2 x^{- \frac{4}{5}}}{5} + \frac{d}{d x} [3]$

Step 1.2.10

Move $x^{- \frac{4}{5}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{2}{5 x^{\frac{4}{5}}} + \frac{d}{d x} [3]$

Step 1.3

Differentiate using the Constant Rule.

Tap for more steps...

Step 1.3.1

Since $3$ is constant with respect to $x$ , the derivative of $3$ with respect to $x$ is $0$ .

$\frac{2}{5 x^{\frac{4}{5}}} + 0$

Step 1.3.2

Add $\frac{2}{5 x^{\frac{4}{5}}}$ and $0$ .

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

Step 2

Find the second derivative.

Tap for more steps...

Step 2.1

Since $\frac{2}{5}$ is constant with respect to $x$ , the derivative of $\frac{2}{5 x^{\frac{4}{5}}}$ with respect to $x$ is $\frac{2}{5} \frac{d}{d x} [\frac{1}{x^{\frac{4}{5}}}]$ .

$\frac{2}{5} \frac{d}{d x} [\frac{1}{x^{\frac{4}{5}}}]$

Step 2.2

Apply basic rules of exponents.

Tap for more steps...

Step 2.2.1

Rewrite $\frac{1}{x^{\frac{4}{5}}}$ as ${(x^{\frac{4}{5}})}^{- 1}$ .

$\frac{2}{5} \frac{d}{d x} [{(x^{\frac{4}{5}})}^{- 1}]$

Step 2.2.2

Multiply the exponents in ${(x^{\frac{4}{5}})}^{- 1}$ .

Tap for more steps...

Step 2.2.2.1

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

$\frac{2}{5} \frac{d}{d x} [x^{\frac{4}{5} \cdot - 1}]$

Step 2.2.2.2

Multiply $\frac{4}{5} \cdot - 1$ .

Tap for more steps...

Step 2.2.2.2.1

Combine $\frac{4}{5}$ and $- 1$ .

$\frac{2}{5} \frac{d}{d x} [x^{\frac{4 \cdot - 1}{5}}]$

Step 2.2.2.2.2

Multiply $4$ by $- 1$ .

$\frac{2}{5} \frac{d}{d x} [x^{\frac{- 4}{5}}]$

Step 2.2.2.3

Move the negative in front of the fraction.

$\frac{2}{5} \frac{d}{d x} [x^{- \frac{4}{5}}]$

Step 2.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = - \frac{4}{5}$ .

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

Step 2.4

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

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

Step 2.5

Combine $- 1$ and $\frac{5}{5}$ .

$\frac{2}{5} (- \frac{4}{5} x^{- \frac{4}{5} + \frac{- 1 \cdot 5}{5}})$

Step 2.6

Combine the numerators over the common denominator.

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

Step 2.7

Simplify the numerator.

Tap for more steps...

Step 2.7.1

Multiply $- 1$ by $5$ .

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

Step 2.7.2

Subtract $5$ from $- 4$ .

$\frac{2}{5} (- \frac{4}{5} x^{\frac{- 9}{5}})$

Step 2.8

Move the negative in front of the fraction.

$\frac{2}{5} (- \frac{4}{5} x^{- \frac{9}{5}})$

Step 2.9

Combine $x^{- \frac{9}{5}}$ and $\frac{4}{5}$ .

$\frac{2}{5} (- \frac{x^{- \frac{9}{5}} \cdot 4}{5})$

Step 2.10

Multiply $\frac{2}{5}$ by $\frac{x^{- \frac{9}{5}} \cdot 4}{5}$ .

$- \frac{2 (x^{- \frac{9}{5}} \cdot 4)}{5 \cdot 5}$

Step 2.11

Multiply.

Tap for more steps...

Step 2.11.1

Multiply $4$ by $2$ .

$- \frac{8 x^{- \frac{9}{5}}}{5 \cdot 5}$

Step 2.11.2

Multiply $5$ by $5$ .

$- \frac{8 x^{- \frac{9}{5}}}{25}$

Step 2.11.3

Move $x^{- \frac{9}{5}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$f'' (x) = - \frac{8}{25 x^{\frac{9}{5}}}$