Expert: Socrates - 2/26/2004

Question
My question regards the different results obtained in textbooks when one finds the derivative of e to the x and one finds the derivative of ln e (or e to the y).

Discussion of problem:

1. derivative of e^x =   e^x times lin e =  e^x times 1 (since lin e = 1)

or the proof goes like this:

2. derivative of e^x =   e^x times   x'    (that is,  e to the x times the derivative of x) =  e^x  times 1

Thus, the textbooks always show that the derivative of  e^x  is e^x.  

However, when it comes to finding the derivative of lin x, the procedure seems to be different.

First, it is pointed out that lin x is the same as e^y =   x.  Typically, the textbooks then go on
to find the derivative of ey as follows:

e^y =   x

so  the derivative of e^y = e^y times lin e times y'   (HERE IS THE PROBLEM, the text books stick in this extra y', sometimes you see this in the form dy/dx)

so

x' = e^y times lin e times y'

and (since the derivative of x is 1)

1 =  e^y times lin e times y'

so (the textbooks always refer to implicit differentiation around about this stage)

y' = 1/ e^y times lin e

and

y' = 1/ x times lin e     and since lin e is 1 we have

y' =  1/x (the derivative of y is 1/x)

and so this proves that the derivative of ln x is 1/x.

It seems to me that the derivative of ex should be ex just as the derivative of e^y is e^y.

Isn't it basically the same thing? What's going on here?  

Answer
Well , your textbook seems to be confusing you on a simple proof. Here is the argument,

Let y = lnx.

Then e^y = x.

Take the derivative of both sides with respect to the variable x. That means we treat y as a function of x. This gives

y' e^y = 1

Because y = lnx , e^y = e^lnx = x

This means we can substitute x for e^y in  y' e^y = 1 .

This gives y' x = 1

Then y' = 1/x , which is the result you want.

You must understand that the derivative of e^y is not e^y when the derivative is taken with respect to x. The derivative of e^y is e^y when the derivative is taken with respect to y.

Have you seen the rule (e^f(x))' = f'(x) e^f(x) ?
That is the rule we are using when we get y'e^y as the derivative of e^y. You must see that y is a function of x and is treated as a function , not as a single , independent variable.

I hope this helps. I would need to see your book to try to explain further.