Calculus Examples

$h (x) = \arccos (\ln (\sqrt{x}))$

Step 1

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

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

Step 2

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

Tap for more steps...

Step 2.1

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

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

Step 2.2

The derivative of $\arccos (u_{1})$ with respect to $u_{1}$ is $- \frac{1}{\sqrt{1 - {u_{1}}^{2}}}$ .

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

Step 2.3

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

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

Step 3

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

Tap for more steps...

Step 3.1

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

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

Step 3.2

The derivative of $\ln (u_{2})$ with respect to $u_{2}$ is $\frac{1}{u_{2}}$ .

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

Step 3.3

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

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

Step 4

Differentiate using the Power Rule.

Tap for more steps...

Step 4.1

Multiply $\frac{1}{x^{\frac{1}{2}}}$ by $\frac{1}{\sqrt{1 - \ln^{2} (x^{\frac{1}{2}})}}$ .

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

Step 4.2

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

$- \frac{1}{x^{\frac{1}{2}} \sqrt{1 - \ln^{2} (x^{\frac{1}{2}})}} (\frac{1}{2} x^{\frac{1}{2} - 1})$

Step 5

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$- \frac{1}{x^{\frac{1}{2}} \sqrt{1 - \ln^{2} (x^{\frac{1}{2}})}} (\frac{1}{2} x^{\frac{1}{2} - 1 \cdot \frac{2}{2}})$

Step 6

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

$- \frac{1}{x^{\frac{1}{2}} \sqrt{1 - \ln^{2} (x^{\frac{1}{2}})}} (\frac{1}{2} x^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}})$

Step 7

Combine the numerators over the common denominator.

$- \frac{1}{x^{\frac{1}{2}} \sqrt{1 - \ln^{2} (x^{\frac{1}{2}})}} (\frac{1}{2} x^{\frac{1 - 1 \cdot 2}{2}})$

Step 8

Simplify the numerator.

Tap for more steps...

Step 8.1

Multiply $- 1$ by $2$ .

$- \frac{1}{x^{\frac{1}{2}} \sqrt{1 - \ln^{2} (x^{\frac{1}{2}})}} (\frac{1}{2} x^{\frac{1 - 2}{2}})$

Step 8.2

Subtract $2$ from $1$ .

$- \frac{1}{x^{\frac{1}{2}} \sqrt{1 - \ln^{2} (x^{\frac{1}{2}})}} (\frac{1}{2} x^{\frac{- 1}{2}})$

Step 9

Move the negative in front of the fraction.

$- \frac{1}{x^{\frac{1}{2}} \sqrt{1 - \ln^{2} (x^{\frac{1}{2}})}} (\frac{1}{2} x^{- \frac{1}{2}})$

Step 10

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

$- \frac{1}{x^{\frac{1}{2}} \sqrt{1 - \ln^{2} (x^{\frac{1}{2}})}} \cdot \frac{x^{- \frac{1}{2}}}{2}$

Step 11

Multiply $\frac{x^{- \frac{1}{2}}}{2}$ by $\frac{1}{x^{\frac{1}{2}} \sqrt{1 - \ln^{2} (x^{\frac{1}{2}})}}$ .

$- \frac{x^{- \frac{1}{2}}}{2 (x^{\frac{1}{2}} \sqrt{1 - \ln^{2} (x^{\frac{1}{2}})})}$

Step 12

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

$- \frac{1}{2 x^{\frac{1}{2}} \sqrt{1 - \ln^{2} (x^{\frac{1}{2}})} x^{\frac{1}{2}}}$

Step 13

Simplify the denominator.

Tap for more steps...

Step 13.1

Multiply $x^{\frac{1}{2}}$ by $x^{\frac{1}{2}}$ by adding the exponents.

Tap for more steps...

Step 13.1.1

Move $x^{\frac{1}{2}}$ .

$- \frac{1}{2 (x^{\frac{1}{2}} x^{\frac{1}{2}}) \sqrt{1 - \ln^{2} (x^{\frac{1}{2}})}}$

Step 13.1.2

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

$- \frac{1}{2 x^{\frac{1}{2} + \frac{1}{2}} \sqrt{1 - \ln^{2} (x^{\frac{1}{2}})}}$

Step 13.1.3

Combine the numerators over the common denominator.

$- \frac{1}{2 x^{\frac{1 + 1}{2}} \sqrt{1 - \ln^{2} (x^{\frac{1}{2}})}}$

Step 13.1.4

Add $1$ and $1$ .

$- \frac{1}{2 x^{\frac{2}{2}} \sqrt{1 - \ln^{2} (x^{\frac{1}{2}})}}$

Step 13.1.5

Divide $2$ by $2$ .

$- \frac{1}{2 x^{1} \sqrt{1 - \ln^{2} (x^{\frac{1}{2}})}}$

Step 13.2

Simplify $2 x^{1} \sqrt{1 - \ln^{2} (x^{\frac{1}{2}})}$ .

$- \frac{1}{2 x \sqrt{1 - \ln^{2} (x^{\frac{1}{2}})}}$