How can you differentiate a function when the variable ( eg x ) is the exponent? Eg F(x)= 2 raised to the x power?

Answer

So the question is, how do you lead the function

f(x) = a^X

af, with a being a strict positive number (different from e, and from 1). So an exponential function with base a. The formula for this is f ‘ (x) = a^X . ln a

This is very easy to find : since a^X always strictly positive you can always take a natural logarithm and then the e-power again. Since the two are each other’s inverse, they cancel each other out. So :

f(x) = exp [  ln ( a^x) ]

So by exp I just mean “e to the power”.
Now you know that a logarithm has the following property : ln a^b = b . ln a

So : f(x) = exp [ x ln(a) ]

The exponential function a^X has thus now been converted into an “ordinary” exponential function.
ln(a) is simply a constant. So now just deduce ;

f'(x) = exp [ x ln(a) ] . ln(a) = exp [  ln(a^x) ] . ln(a) = a^X . ln(a)

Note that if a is the number e, you indeed “just” have e^X finds as a derivative of e^X.
After all, ln(e) = 1

Another important consequence : the derivative of a^f(x)

is thus with the chain rule : a^f(x) . ln(a) . f’ (x)