Calculus Examples

$y = \arcsin (\sqrt{x})$

Step 1

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

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

Step 2

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

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

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

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

Step 3

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

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Step 3.1

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

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

Step 3.2

Cancel the common factor of $2$ .

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Step 3.2.1

Cancel the common factor.

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

Step 3.2.2

Rewrite the expression.

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

Step 4

Simplify.

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

Step 5

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

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

Step 6

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

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

Step 7

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

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

Step 8

Combine the numerators over the common denominator.

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

Step 9

Simplify the numerator.

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Step 9.1

Multiply $- 1$ by $2$ .

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

Step 9.2

Subtract $2$ from $1$ .

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

Step 10

Move the negative in front of the fraction.

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

Step 11

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

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

Step 12

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

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

Step 13

Simplify the expression.

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Step 13.1

Move $2$ to the left of $\sqrt{1 - x}$ .

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

Step 13.2

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

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

Step 13.3

Reorder terms.

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