What is the derivative of e^pi?

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The derivative is ##0##. There are several ways to get this answer.

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I think it is worth pointing out to students that, while there is one correct answer to the question, “What is the derivative of function ##f##?”, there are often several ways to get to that answer.

Any legitimate way of thinking about the function and correctly applying the rules of differentiation will result in the correct answer.

##f(x) = e^pi##

Constant

If we are fortunale enough to recogize that ##e^pi## is a constant (close to ##2.718^3.1416## whatever that number is!), then we can immediately conclude that ##d/dx(e^pi) = 0##

plus

We know that ##d/dx(x^pi) = pix^(pi-1)##. Here we don’t have ##x##, we have instead ##u = g(x) = e##, so we need the chain rule:

##d/dx(u^pi) = piu^(pi-1) (du)/dx##

In this case ##(du)/dx = d/dx(e) = 0##, so we write:

##d/dx(e^pi) = pie^(pi-1)*(0) = 0##

Exponential plus chain

We know that ##d/dx(e^x) = e^x## and ##d/dx(e^u) = e^u (du)/dx##
Looking at it this way, we have ##u = g(x) = pi##, whose derivative is ##0##.

So, in this case we get:

##d/dx(e^pi) = e^pi d/dx(pi) = e^pi *0 = 0##

Logarithmic Differentiation

Let ##y = e^pi##, so that ##lny = pilne##, but ##lne =1##, so we have:

##lny = pi## and differentiation gets us:

##1/y dy/dx = 0##, so

##dy/dx = 0*y = 0e^pi = 0##

But what if i didn’t notice that ##lne = 1##?

It’s ok, when we find the derivative, we’ll use the chain rule (again) ##d/dx(lne) = 1/e d/dx(e) = 1/e *0 = 0##

The point of my answer is not to confuse, but to tell students that any correct application of differentiation formulas will get the correct answer, and to show some of the possible applications of this general principle.



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