What is the derivative of \sqrt[3]{arcsin(x}) ?

The derivative of \sqrt[3]{arcsin(x}) is 1/(3arcsin^{2/3)(x)sqrt(1-x^2)}

What is the first derivative of \sqrt[3]{arcsin(x}) ?

The first derivative of \sqrt[3]{arcsin(x}) is 1/(3arcsin^{2/3)(x)sqrt(1-x^2)}

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

Solution steps

Apply the chain rule:

\frac{1}{3 ( arcsin ( x ) ) ^{\frac{2}{3}}} \frac{d}{d x} (arcsin (x))

= \frac{1}{3 ( arcsin ( x ) ) ^{\frac{2}{3}}} \frac{d}{d x} (arcsin (x))

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

= \frac{1}{3 ( arcsin ( x ) ) ^{\frac{2}{3}}} \cdot \frac{1}{1 - x ^{2}}

Simplify

\frac{1}{3 arcsin ^{\frac{2}{3}} ( x )} \cdot \frac{1}{1 - x ^{2}} : \frac{1}{3 arcsin ^{\frac{2}{3}} ( x ) 1 - x ^{2}}

= \frac{1}{3 arcsin ^{\frac{2}{3}} ( x ) 1 - x ^{2}}

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What is the derivative of \sqrt[3]{arcsin(x}) ?
The derivative of \sqrt[3]{arcsin(x}) is 1/(3arcsin^{2/3)(x)sqrt(1-x^2)}
What is the first derivative of \sqrt[3]{arcsin(x}) ?
The first derivative of \sqrt[3]{arcsin(x}) is 1/(3arcsin^{2/3)(x)sqrt(1-x^2)}