Calculus Examples

$y = \arcsin (\sqrt{2 x})$

Step 1

Simplify with factoring out.

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Step 1.1

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

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

Step 1.2

Factor $2$ out of $2 x$ .

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

Step 1.3

Apply the product rule to $2 (x)$ .

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

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 $g (x) = 2^{\frac{1}{2}} x^{\frac{1}{2}}$ .

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

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

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

Step 3

Differentiate.

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Step 3.1

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

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

Step 3.2

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

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

Step 3.3

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

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

Step 4

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

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

Step 5

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

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

Step 6

Combine the numerators over the common denominator.

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

Step 7

Simplify the numerator.

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Step 7.1

Multiply $- 1$ by $2$ .

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

Step 7.2

Subtract $2$ from $1$ .

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

Step 8

Move the negative in front of the fraction.

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

Step 9

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

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

Step 10

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

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

Step 11

Simplify the expression.

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Step 11.1

Move $2$ to the left of $\sqrt{1 - {(2^{\frac{1}{2}} x^{\frac{1}{2}})}^{2}}$ .

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

Step 11.2

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

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

Step 11.3

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

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

Step 12

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

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Step 12.1

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

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

Step 12.2

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

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Step 12.2.1

Raise $2$ to the power of $1$ .

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

Step 12.2.2

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

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

Step 12.3

Write $1$ as a fraction with a common denominator.

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

Step 12.4

Combine the numerators over the common denominator.

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

Step 12.5

Add $- 1$ and $2$ .

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

Step 13

Simplify.

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Step 13.1

Apply the product rule to $2^{\frac{1}{2}} x^{\frac{1}{2}}$ .

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

Step 13.2

Combine terms.

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Step 13.2.1

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

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Step 13.2.1.1

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

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

Step 13.2.1.2

Cancel the common factor of $2$ .

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Step 13.2.1.2.1

Cancel the common factor.

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

Step 13.2.1.2.2

Rewrite the expression.

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

Step 13.2.2

Evaluate the exponent.

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

Step 13.2.3

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

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Step 13.2.3.1

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

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

Step 13.2.3.2

Cancel the common factor of $2$ .

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Step 13.2.3.2.1

Cancel the common factor.

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

Step 13.2.3.2.2

Rewrite the expression.

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

Step 13.2.4

Simplify.

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

Step 13.2.5

Multiply $2$ by $- 1$ .

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

Step 13.3

Reorder terms.

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

Step 13.4

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

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

Step 13.5

Combine and simplify the denominator.

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Step 13.5.1

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

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

Step 13.5.2

Move $\sqrt{1 - 2 x}$ .

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

Step 13.5.3

Raise $\sqrt{1 - 2 x}$ to the power of $1$ .

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

Step 13.5.4

Raise $\sqrt{1 - 2 x}$ to the power of $1$ .

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

Step 13.5.5

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

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

Step 13.5.6

Add $1$ and $1$ .

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

Step 13.5.7

Rewrite ${\sqrt{1 - 2 x}}^{2}$ as $1 - 2 x$ .

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Step 13.5.7.1

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

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

Step 13.5.7.2

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

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

Step 13.5.7.3

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

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

Step 13.5.7.4

Cancel the common factor of $2$ .

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Step 13.5.7.4.1

Cancel the common factor.

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

Step 13.5.7.4.2

Rewrite the expression.

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

Step 13.5.7.5

Simplify.

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