Find derivative of 5^(arcsine(x))

1. Jan 9, 2008

oceanflavored

1. The problem statement, all variables and given/known data
how do you take the derivative of y = 5^(inverse sine of x)?

2. Relevant equations
the formula to take the derivative of arcsin x = 1 / square root of 1 - X2

3. 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!

thanks bunches :)

2. Jan 9, 2008

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.

3. Jan 10, 2008

HallsofIvy

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