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

In summary, the conversation is discussing a problem involving differentiation of a natural log function. The solution involves splitting the differentiation across the terms and using the property that (1/2)^c can be rewritten as 2^-c. There should be a minus sign in the final answer.
  • #1
Pyroadept
89
0

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 :)
 
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  • #2
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?
 
  • #3
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 = eln a, for a > 0, and a != 1,
then ax = (eln a)x = ex ln a.
 
  • #4
Pyroadept said:
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.
 

1. What is the formula for differentiating a^x with regard to x?

The formula for differentiating a^x with regard to x is d/dx(a^x) = a^x * ln(a).

2. Can you explain the meaning of each component in the differentiation formula?

The letter a represents the base of the exponential function, while x is the variable with respect to which we are differentiating. The ln(a) represents the natural logarithm of the base a.

3. How do you differentiate a^x with regard to x when a is a constant?

If a is a constant in the exponential function, then the derivative of a^x with regard to x is simply a^x * ln(a).

4. How do you differentiate a^x with regard to x when a is a variable?

If a is a variable in the exponential function, then the derivative of a^x with regard to x is a^x * ln(a) * d/dx(a).

5. Can you provide an example of differentiating a^x with regard to x?

Sure, let's say we have the function f(x) = 2^x. The derivative of f(x) with regard to x would be d/dx(2^x) = 2^x * ln(2) * d/dx(2) = 2^x * ln(2) * 0 = 2^x * ln(2) * 0 = 0. Therefore, the derivative of 2^x with regard to x is 0.

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