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# The Derivatives of e^x and ln(x)

What you need to know is that the student work is the stuff typed in red, and that this came on a take-home quiz:

Why does this student think that ln(e) = x^1. Or did I get that wrong? Are there ideas that “feel similar” that he’s confusing, or is it something else?

(Thanks Taylor!)

## 5 replies on “The Derivatives of e^x and ln(x)”

What is frustrating to me looking at this is the fact that the student does not seem to recognize that e^x and a^x are structurally the same thing. It seems pretty clear that e is seen as an entirely different kind of thing. This carries over to the ln versus log base a question as well. It feels to me that this student is just trying to sort and memorize a whole bunch of facts rather than seeing any connections here. In the first question if the x^1 had been switched to 1^x we would have had a correct answer for an incorrect reason – these are always hard to navigate as a teacher.
I think I would start by asking the student to look at a variety of a^x functions and emphasize the correct answer here, then bridge to something like pi^x before looking at the e^x. Try to convince the student of the stability of the pattern there. Perhaps correcting this would lead naturally to correcting the log questions.

It is curious to me that the derivative of a^x is correct, but somehow incorrect for e^x. I’ve never remembered the general formula for d/dx(a^x), instead using d/dx(e^x) = e^x and then using a transformation and chain rule: a^x = e^(ln(a)*x).

It is possible that the student looked up answers for a^x and ln(a) but tried those with a=e on his/her own.

Laura C.says:

I can definitely see this happening. Students have an unfortunate tendency to confuse x^1 and the number 1.

Since ln (e) = 1, transferring from the general case to a = e would give you:

e^x * ln(e) = e^x * 1 = e^x

But just judging by the number of students to think that x^0 = x^1 instead of just 1, I’d say something similar is probably happening here. Instead of ln(e) = 1, the student has decided that ln(e) = x^1.

Perhaps a failed chain rule? That is to say, a student who knows d/dx e^{x^2} = (2x) e^{x^2}, d/dx e^{x^3} = (3x^2) e^{x^3} and extrapolates wrongly to d/dx e^x = (1 x) e^x (when it should be 1 x^0 e^x.)