What is derivative of f(x)=e^(arcsin x)?

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sciencesolve | Teacher | (Level 3) Educator Emeritus

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You need to use logarithmic differentiation, hence, taking common logarithms both sides, yields:

`f(x) = y => y = e^(arcsin x) => ln y = ln e^(arcsin x)`

Using the power property of logarithms, yields:

`ln y = arcsin x*ln e => ln y = arcsin x*1 => ln y = arcsin x`

Differentiating both sides yields:

`1/y*y' = 1/(sqrt(1 - x^2))`

`y' = y*(1/(sqrt(1 - x^2)))`

Replacing e^(arcsin x) for y, yields:

`y' = (e^(arcsin x))/(sqrt(1 - x^2))`

Hence, evaluating the derivative of the given function, using logarithmic differentiation, yields `f'(x) = (e^(arcsin x))/(sqrt(1 - x^2)).`

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