Calculus Examples

$\ln (\sqrt[5]{x^{2} + 1})$

Step 1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[5]{x^{2} + 1}$ as ${(x^{2} + 1)}^{\frac{1}{5}}$ .

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

Step 2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \ln (x)$ and $g (x) = {(x^{2} + 1)}^{\frac{1}{5}}$ .

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Step 2.1

To apply the Chain Rule, set $u_{1}$ as ${(x^{2} + 1)}^{\frac{1}{5}}$ .

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

Step 2.2

The derivative of $\ln (u_{1})$ with respect to $u_{1}$ is $\frac{1}{u_{1}}$ .

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

Step 2.3

Replace all occurrences of $u_{1}$ with ${(x^{2} + 1)}^{\frac{1}{5}}$ .

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

Step 3

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{\frac{1}{5}}$ and $g (x) = x^{2} + 1$ .

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Step 3.1

To apply the Chain Rule, set $u_{2}$ as $x^{2} + 1$ .

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

Step 3.2

Differentiate using the Power Rule which states that $\frac{d}{d u_{2}} [{u_{2}}^{n}]$ is $n {u_{2}}^{n - 1}$ where $n = \frac{1}{5}$ .

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

Step 3.3

Replace all occurrences of $u_{2}$ with $x^{2} + 1$ .

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

Step 4

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

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

Step 5

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

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

Step 6

Combine the numerators over the common denominator.

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

Step 7

Simplify the numerator.

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Step 7.1

Multiply $- 1$ by $5$ .

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

Step 7.2

Subtract $5$ from $1$ .

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

Step 8

Move the negative in front of the fraction.

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

Step 9

Combine $\frac{1}{5}$ and ${(x^{2} + 1)}^{- \frac{4}{5}}$ .

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

Step 10

Move ${(x^{2} + 1)}^{- \frac{4}{5}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 11

Multiply $\frac{1}{5 {(x^{2} + 1)}^{\frac{4}{5}}}$ by $\frac{1}{{(x^{2} + 1)}^{\frac{1}{5}}}$ .

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

Step 12

Multiply ${(x^{2} + 1)}^{\frac{4}{5}}$ by ${(x^{2} + 1)}^{\frac{1}{5}}$ by adding the exponents.

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Step 12.1

Move ${(x^{2} + 1)}^{\frac{1}{5}}$ .

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

Step 12.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 12.3

Combine the numerators over the common denominator.

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

Step 12.4

Add $1$ and $4$ .

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

Step 12.5

Divide $5$ by $5$ .

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

Step 13

Simplify $5 {(x^{2} + 1)}^{1}$ .

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

Step 14

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

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

Step 15

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

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

Step 16

Since $1$ is constant with respect to $x$ , the derivative of $1$ with respect to $x$ is $0$ .

$\frac{1}{5 (x^{2} + 1)} (2 x + 0)$

Step 17

Combine fractions.

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Step 17.1

Add $2 x$ and $0$ .

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

Step 17.2

Combine $2$ and $\frac{1}{5 (x^{2} + 1)}$ .

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

Step 17.3

Combine $\frac{2}{5 (x^{2} + 1)}$ and $x$ .

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