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**1. The problem statement, all variables and given/known data**

In the last couple threads, it has become apparent that I need to organize my understanding of some of the derivative rules, specifically as they relate to exponential functions, such as e^x.

So I wrote out a couple possible ways of evaluating e^x. Could you tell me if these are valid or invalid methods.

I want to say that they are all correct methods except for the last method. Even though I get the correct answer in the last (4th) attempt, it does not work for any other exponential function, thus is coincident that it works with e^x.

So basically, in terms of functions in which the variable is in the exponent, the methods of exponential rule, chain rule, or taking ln of both sides are valid. However the power rule can not be used. (and of course the inaccurate method of using the chain rule as in attempt #4 also can not be used)

Is this an accurate assessment of taking derivatives of exponential functions? Is there anything i'm missing or anything i'm mixing up?

**2. Relevant equations**

**3. The attempt at a solution**