Question

How do you find the derivative of `2^arcsin(x)`?

Answer 1

`"d"/("d"x) 2^{arcsin(x)} = (ln(2)2^{arcsin(x)})/(sqrt(1-x^2))`.

Explanation:

Rewrite `2^arcsin(x)=e^{ln(2)arcsin(x)}`.

Then, by the chain rule,

`"d"/("d"x) 2^{arcsin(x)} = ln(2) "d"/("d"x) (arcsin(x)) e^{ln(2)arcsin(x)}`.

The standard result for the derivative of `arcsin(x)` is `"d"/("d"x) (arcsin(x)) = 1/sqrt(1-x^2)`.

Then we see that,

`"d"/("d"x) 2^{arcsin(x)} = (ln(2)2^{arcsin(x)})/(sqrt(1-x^2))`.