Algebra Examples

$\arcsin (\sqrt{\sin (11 x)})$

Step 1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{\sin (11 x)}$ as ${\sin (11 x)}^{\frac{1}{2}}$ .

$\frac{d}{d x} [\arcsin ({\sin (11 x)}^{\frac{1}{2}})]$

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 $g (x) = {\sin (11 x)}^{\frac{1}{2}}$ .

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Step 2.1

To apply the Chain Rule, set $u_{1}$ as ${\sin (11 x)}^{\frac{1}{2}}$ .

$\frac{d}{d u_{1}} [\arcsin (u_{1})] \frac{d}{d x} [{\sin (11 x)}^{\frac{1}{2}}]$

Step 2.2

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

$\frac{1}{\sqrt{1 - {u_{1}}^{2}}} \frac{d}{d x} [{\sin (11 x)}^{\frac{1}{2}}]$

Step 2.3

Replace all occurrences of $u_{1}$ with ${\sin (11 x)}^{\frac{1}{2}}$ .

$\frac{1}{\sqrt{1 - {({\sin (11 x)}^{\frac{1}{2}})}^{2}}} \frac{d}{d x} [{\sin (11 x)}^{\frac{1}{2}}]$

Step 3

Multiply the exponents in ${({\sin (11 x)}^{\frac{1}{2}})}^{2}$ .

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Step 3.1

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

$\frac{1}{\sqrt{1 - {\sin (11 x)}^{\frac{1}{2} \cdot 2}}} \frac{d}{d x} [{\sin (11 x)}^{\frac{1}{2}}]$

Step 3.2

Cancel the common factor of $2$ .

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Step 3.2.1

Cancel the common factor.

$\frac{1}{\sqrt{1 - {\sin (11 x)}^{\frac{1}{2} \cdot 2}}} \frac{d}{d x} [{\sin (11 x)}^{\frac{1}{2}}]$

Step 3.2.2

Rewrite the expression.

$\frac{1}{\sqrt{1 - \sin^{1} (11 x)}} \frac{d}{d x} [{\sin (11 x)}^{\frac{1}{2}}]$

Step 4

Simplify.

$\frac{1}{\sqrt{1 - \sin (11 x)}} \frac{d}{d x} [{\sin (11 x)}^{\frac{1}{2}}]$

Step 5

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{\frac{1}{2}}$ and $g (x) = \sin (11 x)$ .

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Step 5.1

To apply the Chain Rule, set $u_{2}$ as $\sin (11 x)$ .

$\frac{1}{\sqrt{1 - \sin (11 x)}} (\frac{d}{d u_{2}} [{u_{2}}^{\frac{1}{2}}] \frac{d}{d x} [\sin (11 x)])$

Step 5.2

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

$\frac{1}{\sqrt{1 - \sin (11 x)}} (\frac{1}{2} {u_{2}}^{\frac{1}{2} - 1} \frac{d}{d x} [\sin (11 x)])$

Step 5.3

Replace all occurrences of $u_{2}$ with $\sin (11 x)$ .

$\frac{1}{\sqrt{1 - \sin (11 x)}} (\frac{1}{2} {\sin (11 x)}^{\frac{1}{2} - 1} \frac{d}{d x} [\sin (11 x)])$

Step 6

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

$\frac{1}{\sqrt{1 - \sin (11 x)}} (\frac{1}{2} {\sin (11 x)}^{\frac{1}{2} - 1 \cdot \frac{2}{2}} \frac{d}{d x} [\sin (11 x)])$

Step 7

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

$\frac{1}{\sqrt{1 - \sin (11 x)}} (\frac{1}{2} {\sin (11 x)}^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}} \frac{d}{d x} [\sin (11 x)])$

Step 8

Combine the numerators over the common denominator.

$\frac{1}{\sqrt{1 - \sin (11 x)}} (\frac{1}{2} {\sin (11 x)}^{\frac{1 - 1 \cdot 2}{2}} \frac{d}{d x} [\sin (11 x)])$

Step 9

Simplify the numerator.

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Step 9.1

Multiply $- 1$ by $2$ .

$\frac{1}{\sqrt{1 - \sin (11 x)}} (\frac{1}{2} {\sin (11 x)}^{\frac{1 - 2}{2}} \frac{d}{d x} [\sin (11 x)])$

Step 9.2

Subtract $2$ from $1$ .

$\frac{1}{\sqrt{1 - \sin (11 x)}} (\frac{1}{2} {\sin (11 x)}^{\frac{- 1}{2}} \frac{d}{d x} [\sin (11 x)])$

Step 10

Combine fractions.

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Step 10.1

Move the negative in front of the fraction.

$\frac{1}{\sqrt{1 - \sin (11 x)}} (\frac{1}{2} {\sin (11 x)}^{- \frac{1}{2}} \frac{d}{d x} [\sin (11 x)])$

Step 10.2

Combine $\frac{1}{2}$ and ${\sin (11 x)}^{- \frac{1}{2}}$ .

$\frac{1}{\sqrt{1 - \sin (11 x)}} (\frac{{\sin (11 x)}^{- \frac{1}{2}}}{2} \frac{d}{d x} [\sin (11 x)])$

Step 10.3

Move ${\sin (11 x)}^{- \frac{1}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{1}{\sqrt{1 - \sin (11 x)}} (\frac{1}{2 {\sin (11 x)}^{\frac{1}{2}}} \frac{d}{d x} [\sin (11 x)])$

Step 10.4

Multiply $\frac{1}{2 {\sin (11 x)}^{\frac{1}{2}}}$ by $\frac{1}{\sqrt{1 - \sin (11 x)}}$ .

$\frac{1}{2 {\sin (11 x)}^{\frac{1}{2}} \sqrt{1 - \sin (11 x)}} \frac{d}{d x} [\sin (11 x)]$

Step 11

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \sin (x)$ and $g (x) = 11 x$ .

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Step 11.1

To apply the Chain Rule, set $u_{3}$ as $11 x$ .

$\frac{1}{2 {\sin (11 x)}^{\frac{1}{2}} \sqrt{1 - \sin (11 x)}} (\frac{d}{d u_{3}} [\sin (u_{3})] \frac{d}{d x} [11 x])$

Step 11.2

The derivative of $\sin (u_{3})$ with respect to $u_{3}$ is $\cos (u_{3})$ .

$\frac{1}{2 {\sin (11 x)}^{\frac{1}{2}} \sqrt{1 - \sin (11 x)}} (\cos (u_{3}) \frac{d}{d x} [11 x])$

Step 11.3

Replace all occurrences of $u_{3}$ with $11 x$ .

$\frac{1}{2 {\sin (11 x)}^{\frac{1}{2}} \sqrt{1 - \sin (11 x)}} (\cos (11 x) \frac{d}{d x} [11 x])$

Step 12

Differentiate.

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Step 12.1

Combine $\cos (11 x)$ and $\frac{1}{2 {\sin (11 x)}^{\frac{1}{2}} \sqrt{1 - \sin (11 x)}}$ .

$\frac{\cos (11 x)}{2 {\sin (11 x)}^{\frac{1}{2}} \sqrt{1 - \sin (11 x)}} \frac{d}{d x} [11 x]$

Step 12.2

Since $11$ is constant with respect to $x$ , the derivative of $11 x$ with respect to $x$ is $11 \frac{d}{d x} [x]$ .

$\frac{\cos (11 x)}{2 {\sin (11 x)}^{\frac{1}{2}} \sqrt{1 - \sin (11 x)}} (11 \frac{d}{d x} [x])$

Step 12.3

Combine $11$ and $\frac{\cos (11 x)}{2 {\sin (11 x)}^{\frac{1}{2}} \sqrt{1 - \sin (11 x)}}$ .

$\frac{11 \cos (11 x)}{2 {\sin (11 x)}^{\frac{1}{2}} \sqrt{1 - \sin (11 x)}} \frac{d}{d x} [x]$

Step 12.4

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

$\frac{11 \cos (11 x)}{2 {\sin (11 x)}^{\frac{1}{2}} \sqrt{1 - \sin (11 x)}} \cdot 1$

Step 12.5

Multiply $\frac{11 \cos (11 x)}{2 {\sin (11 x)}^{\frac{1}{2}} \sqrt{1 - \sin (11 x)}}$ by $1$ .

$\frac{11 \cos (11 x)}{2 {\sin (11 x)}^{\frac{1}{2}} \sqrt{1 - \sin (11 x)}}$