Question

Find the derivative of the function. Find the domains of the function and its derivative.
f(x)=arcsin (e^x)

I already know that the derivative is (e^x)/(sqrt(1-e^2x) I'm mostly having trouble find the domains of both functions.

Answer 1

Try breaking it up into smaller pieces and see if you can assemble it back together. What is the domain of arcsine and e^x?

Answer 2

The domain of arcsin is -1<x<1
The domain of e^x is all real numbers???

Answer 3

I 'm not really sure where to go from there.

Answer 4

Let's identify our goal first. We want to know the domain of our function. What does that mean? It means we need to find what values of x we can plug in. So how do we determine that? Let's start by looking at it from the outside in. We know we can pick any value from -1 to 1 inside plain arcsine. So now we can effectively think of what is the possible range we can pick for e^x? For instance if we pick x=3 we will have arcsin(e^3) and since e^3 > 1 we know it won't be in our domain! So we should set e^x=1 since that's the maximum value in our domain of plain arcsine. Now we can solve for the upper bound on the actual domain we want. Then we can also do that for the lower bound.

Answer 5

so we could plug in e^x into the arcsine domain and get -1<e^x < 1
solve and I would get -infinity<x<0 but what about the domain of the derivitive?

Answer 6

Any ideas? Remember, one of the two most popular things that restrict your domain are square roots of negative numbers and dividing by zero.

Answer 7

Also good job on finding the domain of the first function. =)

Answer 8

so set the sqrt > 0

sqrt(1-e^2x)>0
1-e^2x>0
-e^2x>-1
e^2x<1
2x<0 (ln1=0)
x<0

is that it???

Answer 9

and thanks!