What is the derivative of e^pi?

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The derivative is ##0##. There are various ways to get this rejoinder.

I hold it is value pointing out to students that, while there is one chasten rejoinder to the investigation, "What is the derivative of employment ##f##?", there are repeatedly various ways to get to that rejoinder.

Any allowable way of holding environing the employment and chastenly adduceing the governments of differentiation procure consequence in the chasten rejoinder.

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

Constant

If we are fortunale abundance to recogize that ##e^pi## is a firm (halt to ##2.718^3.1416## whatever that number is!), then we can instantly decide that ##d/dx(e^pi) = 0##

plus

We distinguish that ##d/dx(x^pi) = pix^(pi-1)##. Here we don't possess ##x##, we possess instead ##u = g(x) = e##, so we want the tie government:

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

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

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

Exponential plus tie

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

So, in this circumstance 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 possess:

##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 mention that ##lne = 1##?

It's ok, when we experience the derivative, we'll use the tie government (again) ##d/dx(lne) = 1/e d/dx(e) = 1/e *0 = 0##

The point of my rejoinder is not to disorder, but to divulge students that any chasten impression of differentiation formulas procure get the chasten rejoinder, and to pretence some of the practicable impressions of this public energy.


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