Find derivative of 5^(arcsine(x))

[oceanflavored]

Homework Statement

how do you take the derivative of y = 5^(inverse sine of x)?

Homework Equations

the formula to take the derivative of arcsin x = 1 / square root of 1 - X2

The Attempt at a Solution

i tried to use the chain rule so that i did...

5 ^ (inverse sin of x) times (the derivative formula)

but i don't think it's right!

PLEASE HELP!

thanks bunches :)

 

[Defennder]

You have to use implicit differentiation. Firstly you have to "bring the arcsin x down" by taking the natural logarithm of both sides. Then it becomes

$$ln|y|=sin^{-1}x(ln5)$$

Then simply differentiate implicitly.

 

[HallsofIvy]

That is certainly one way to do it and probably simplest but it is not necessary to use "logarithmic differentiation", just the chain rule.

The derivative of a^x, with respect to x, for any positive number a, is ln(a)a^{x}. The derivative of
[tex]a^{sin^{-1}(x)}[/itex] <br /> is, of course,<br /> $$ln(a) a^{sin^{-1}(x)} \frac{d}{dx}sin^{-1}(x)$$[/tex]