Calculus Examples

$y = \arcsin (\frac{1}{\sqrt{x}})$

Step 1

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

$y = \arcsin (\frac{1}{x^{\frac{1}{2}}})$

Step 2

Differentiate both sides of the equation.

$\frac{d}{d x} (y) = \frac{d}{d x} (\arcsin (\frac{1}{x^{\frac{1}{2}}}))$

Step 3

The derivative of $y$ with respect to $x$ is $y'$ .

$y'$

Step 4

Differentiate the right side of the equation.

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Step 4.1

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

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Step 4.1.1

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

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

Step 4.1.2

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

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

Step 4.1.3

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

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

Step 4.2

Differentiate using the Power Rule.

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Step 4.2.1

Rewrite $\frac{1}{x^{\frac{1}{2}}}$ as ${(x^{\frac{1}{2}})}^{- 1}$ .

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

Step 4.2.2

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

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Step 4.2.2.1

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

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

Step 4.2.2.2

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

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

Step 4.2.2.3

Move the negative in front of the fraction.

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

Step 4.2.3

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

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

Step 4.3

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

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

Step 4.4

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

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

Step 4.5

Combine the numerators over the common denominator.

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

Step 4.6

Simplify the numerator.

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Step 4.6.1

Multiply $- 1$ by $2$ .

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

Step 4.6.2

Subtract $2$ from $- 1$ .

$\frac{1}{\sqrt{1 - {(\frac{1}{x^{\frac{1}{2}}})}^{2}}} (- \frac{1}{2} x^{\frac{- 3}{2}})$

Step 4.7

Move the negative in front of the fraction.

$\frac{1}{\sqrt{1 - {(\frac{1}{x^{\frac{1}{2}}})}^{2}}} (- \frac{1}{2} x^{- \frac{3}{2}})$

Step 4.8

Combine $x^{- \frac{3}{2}}$ and $\frac{1}{2}$ .

$\frac{1}{\sqrt{1 - {(\frac{1}{x^{\frac{1}{2}}})}^{2}}} (- \frac{x^{- \frac{3}{2}}}{2})$

Step 4.9

Multiply $\frac{1}{\sqrt{1 - {(\frac{1}{x^{\frac{1}{2}}})}^{2}}}$ by $\frac{x^{- \frac{3}{2}}}{2}$ .

$- \frac{x^{- \frac{3}{2}}}{\sqrt{1 - {(\frac{1}{x^{\frac{1}{2}}})}^{2}} \cdot 2}$

Step 4.10

Simplify the expression.

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Step 4.10.1

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

$- \frac{x^{- \frac{3}{2}}}{2 \cdot \sqrt{1 - {(\frac{1}{x^{\frac{1}{2}}})}^{2}}}$

Step 4.10.2

Move $x^{- \frac{3}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$- \frac{1}{2 \sqrt{1 - {(\frac{1}{x^{\frac{1}{2}}})}^{2}} x^{\frac{3}{2}}}$

Step 4.11

Simplify.

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Step 4.11.1

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

$- \frac{1}{2 \sqrt{1 - \frac{1^{2}}{{(x^{\frac{1}{2}})}^{2}}} x^{\frac{3}{2}}}$

Step 4.11.2

Combine terms.

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Step 4.11.2.1

One to any power is one.

$- \frac{1}{2 \sqrt{1 - \frac{1}{{(x^{\frac{1}{2}})}^{2}}} x^{\frac{3}{2}}}$

Step 4.11.2.2

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

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Step 4.11.2.2.1

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

$- \frac{1}{2 \sqrt{1 - \frac{1}{x^{\frac{1}{2} \cdot 2}}} x^{\frac{3}{2}}}$

Step 4.11.2.2.2

Cancel the common factor of $2$ .

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Step 4.11.2.2.2.1

Cancel the common factor.

$- \frac{1}{2 \sqrt{1 - \frac{1}{x^{\frac{1}{2} \cdot 2}}} x^{\frac{3}{2}}}$

Step 4.11.2.2.2.2

Rewrite the expression.

$- \frac{1}{2 \sqrt{1 - \frac{1}{x^{1}}} x^{\frac{3}{2}}}$

Step 4.11.2.3

Simplify.

$- \frac{1}{2 \sqrt{1 - \frac{1}{x}} x^{\frac{3}{2}}}$

Step 4.11.3

Simplify the denominator.

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Step 4.11.3.1

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

$- \frac{1}{2 \sqrt{\frac{x}{x} - \frac{1}{x}} x^{\frac{3}{2}}}$

Step 4.11.3.2

Combine the numerators over the common denominator.

$- \frac{1}{2 \sqrt{\frac{x - 1}{x}} x^{\frac{3}{2}}}$

Step 4.11.3.3

Rewrite $\sqrt{\frac{x - 1}{x}}$ as $\frac{\sqrt{x - 1}}{\sqrt{x}}$ .

$- \frac{1}{2 \frac{\sqrt{x - 1}}{\sqrt{x}} x^{\frac{3}{2}}}$

Step 4.11.3.4

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

$- \frac{1}{2 (\frac{\sqrt{x - 1}}{\sqrt{x}} \cdot \frac{\sqrt{x}}{\sqrt{x}}) x^{\frac{3}{2}}}$

Step 4.11.3.5

Combine and simplify the denominator.

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Step 4.11.3.5.1

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

$- \frac{1}{2 \frac{\sqrt{x - 1} \sqrt{x}}{\sqrt{x} \sqrt{x}} x^{\frac{3}{2}}}$

Step 4.11.3.5.2

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

$- \frac{1}{2 \frac{\sqrt{x - 1} \sqrt{x}}{{\sqrt{x}}^{1} \sqrt{x}} x^{\frac{3}{2}}}$

Step 4.11.3.5.3

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

$- \frac{1}{2 \frac{\sqrt{x - 1} \sqrt{x}}{{\sqrt{x}}^{1} {\sqrt{x}}^{1}} x^{\frac{3}{2}}}$

Step 4.11.3.5.4

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

$- \frac{1}{2 \frac{\sqrt{x - 1} \sqrt{x}}{{\sqrt{x}}^{1 + 1}} x^{\frac{3}{2}}}$

Step 4.11.3.5.5

Add $1$ and $1$ .

$- \frac{1}{2 \frac{\sqrt{x - 1} \sqrt{x}}{{\sqrt{x}}^{2}} x^{\frac{3}{2}}}$

Step 4.11.3.5.6

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

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Step 4.11.3.5.6.1

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

$- \frac{1}{2 \frac{\sqrt{x - 1} \sqrt{x}}{{(x^{\frac{1}{2}})}^{2}} x^{\frac{3}{2}}}$

Step 4.11.3.5.6.2

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

$- \frac{1}{2 \frac{\sqrt{x - 1} \sqrt{x}}{x^{\frac{1}{2} \cdot 2}} x^{\frac{3}{2}}}$

Step 4.11.3.5.6.3

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

$- \frac{1}{2 \frac{\sqrt{x - 1} \sqrt{x}}{x^{\frac{2}{2}}} x^{\frac{3}{2}}}$

Step 4.11.3.5.6.4

Cancel the common factor of $2$ .

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Step 4.11.3.5.6.4.1

Cancel the common factor.

$- \frac{1}{2 \frac{\sqrt{x - 1} \sqrt{x}}{x^{\frac{2}{2}}} x^{\frac{3}{2}}}$

Step 4.11.3.5.6.4.2

Rewrite the expression.

$- \frac{1}{2 \frac{\sqrt{x - 1} \sqrt{x}}{x^{1}} x^{\frac{3}{2}}}$

Step 4.11.3.5.6.5

Simplify.

$- \frac{1}{2 \frac{\sqrt{x - 1} \sqrt{x}}{x} x^{\frac{3}{2}}}$

Step 4.11.3.6

Combine using the product rule for radicals.

$- \frac{1}{2 \frac{\sqrt{(x - 1) x}}{x} x^{\frac{3}{2}}}$

Step 4.11.3.7

Combine exponents.

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Step 4.11.3.7.1

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

$- \frac{1}{\frac{2 \sqrt{(x - 1) x}}{x} x^{\frac{3}{2}}}$

Step 4.11.3.7.2

Combine $\frac{2 \sqrt{(x - 1) x}}{x}$ and $x^{\frac{3}{2}}$ .

$- \frac{1}{\frac{2 \sqrt{(x - 1) x} x^{\frac{3}{2}}}{x}}$

Step 4.11.3.8

Reduce the expression by cancelling the common factors.

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Step 4.11.3.8.1

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

$- \frac{1}{2 \sqrt{(x - 1) x} x^{\frac{3}{2}} x^{- 1}}$

Step 4.11.3.8.2

Multiply $x^{\frac{3}{2}}$ by $x^{- 1}$ by adding the exponents.

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Step 4.11.3.8.2.1

Move $x^{- 1}$ .

$- \frac{1}{2 \sqrt{(x - 1) x} (x^{- 1} x^{\frac{3}{2}})}$

Step 4.11.3.8.2.2

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

$- \frac{1}{2 \sqrt{(x - 1) x} x^{- 1 + \frac{3}{2}}}$

Step 4.11.3.8.2.3

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

$- \frac{1}{2 \sqrt{(x - 1) x} x^{- 1 \cdot \frac{2}{2} + \frac{3}{2}}}$

Step 4.11.3.8.2.4

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

$- \frac{1}{2 \sqrt{(x - 1) x} x^{\frac{- 1 \cdot 2}{2} + \frac{3}{2}}}$

Step 4.11.3.8.2.5

Combine the numerators over the common denominator.

$- \frac{1}{2 \sqrt{(x - 1) x} x^{\frac{- 1 \cdot 2 + 3}{2}}}$

Step 4.11.3.8.2.6

Simplify the numerator.

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Step 4.11.3.8.2.6.1

Multiply $- 1$ by $2$ .

$- \frac{1}{2 \sqrt{(x - 1) x} x^{\frac{- 2 + 3}{2}}}$

Step 4.11.3.8.2.6.2

Add $- 2$ and $3$ .

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

Step 4.11.4

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

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

Step 4.11.5

Combine and simplify the denominator.

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Step 4.11.5.1

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

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

Step 4.11.5.2

Move $\sqrt{(x - 1) x}$ .

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

Step 4.11.5.3

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

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

Step 4.11.5.4

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

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

Step 4.11.5.5

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

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

Step 4.11.5.6

Add $1$ and $1$ .

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

Step 4.11.5.7

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

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Step 4.11.5.7.1

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

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

Step 4.11.5.7.2

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

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

Step 4.11.5.7.3

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

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

Step 4.11.5.7.4

Cancel the common factor of $2$ .

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Step 4.11.5.7.4.1

Cancel the common factor.

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

Step 4.11.5.7.4.2

Rewrite the expression.

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

Step 4.11.5.7.5

Simplify.

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

Step 4.11.6

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

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Step 4.11.6.1

Move $x$ .

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

Step 4.11.6.2

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

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Step 4.11.6.2.1

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

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

Step 4.11.6.2.2

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

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

Step 4.11.6.3

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

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

Step 4.11.6.4

Combine the numerators over the common denominator.

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

Step 4.11.6.5

Add $2$ and $1$ .

$- \frac{\sqrt{(x - 1) x}}{2 x^{\frac{3}{2}} (x - 1)}$

Step 4.11.7

Reorder factors in $- \frac{\sqrt{(x - 1) x}}{2 x^{\frac{3}{2}} (x - 1)}$ .

$- \frac{\sqrt{x (x - 1)}}{2 x^{\frac{3}{2}} (x - 1)}$

Step 5

Reform the equation by setting the left side equal to the right side.

$y' = - \frac{\sqrt{x (x - 1)}}{2 x^{\frac{3}{2}} (x - 1)}$

Step 6

Replace $y'$ with $\frac{d y}{d x}$ .

$\frac{d y}{d x} = - \frac{\sqrt{x (x - 1)}}{2 x^{\frac{3}{2}} (x - 1)}$