How do you differentiate a^x with regard to x?

[Pyroadept]

Homework Statement

Hi, this is part of a stats problem, in the solutions they go from:

d/dc of ln(1-0.5^c)

then next line they have:

ln(0.5).[0.5^c]/(1-0.5^c)

I don't understand how they did this!

Homework Equations

The Attempt at a Solution

So, I know that the (1-0.5^c) on the bottom comes from differentiating the natural log, and so then the top line must come from differentiating the argument, 1-0.5^c. But I don't see how they did this. I've tried to use exponentials and logs to 'bring down' the power but haven't managed to do it so far.
Could someone please point me in the right direction?

Thanks :)

 

[Pyroadept]

Actually I think I figured it out, split the d/dc across thd 1 and the 0.5^c, but then shouldn't there be a minus sign in the answer?

 

[Mark44]

Remember that (1/2)^c = 2^-c.

When you differentiate 2^-c or any exponential function in a base other than e, the best thing to do is to write the exponential using base e.

Since a = e^{ln a}, for a > 0, and a != 1,
then a^x = (e^{ln a})^x = e^{x ln a}.

 

[Dick]

Actually I think I figured it out, split the d/dc across thd 1 and the 0.5^c, but then shouldn't there be a minus sign in the answer?

Yes, there should be a minus sign in there.