2^x derivitive

hackensack

I need to find the derivitive of y=2^x using the definition of derivitive.

Hurkyl

What have you done so far? What does the definition of derivative say?

HallsofIvy

This was also posted in the calculus section and there are about 10 replies there.

mathwonk

I doubt this is a suitable problem for a novice. even showing convergence is tough. i will look at the other posted answers. there is a good reason people start from the integral definition of ln(x) to derive this result.

HallsofIvy

If f(x)= a^x, the f(x+h)= a^a+x= a^xa^h so
f(a+ h)- f(a)= a^x(a^h- 1).

The derivative is lim (f(x+h)- f(x))/h= a^xlim {(a^h-1)/h}. Notice that that is a^x time a limit that is independent of x. That is, as long as the derivative exists, it is a^x times a constant. The problem is showing that the lim{(a^h-1)/h} EXISTS! And then showing that, if a= 2, that limit is ln(2).

Showing that that limit exists is sufficiently non-trivial that many people (myself included), as mathwonk said, prefer to define ln(x) as the integral, from 1 to x of (1/t)dt. From that, it is possible to prove all properties of ln(x) including (trivially) that the derivative is 1/x. Defining e^x as the inverse function of ln(x) leads to all the properties of e^x (including the fact that it is some number to a power!), in particular that its derivative is e^x itself and, from that, that the derivative of a^x is (ln a) a^x.

quantitative

dunno if i'm missing the point here but...

write
y=2^x
as
y=exp(x.ln2)
=>
y'=ln2.exp(x.ln2)

cepheid

No, you did exactly what HallsofIvy was advocating, he was just pointing out that the question asked for it to be solved using the definition of a derivative, which makes things much harder. Easier to approach things from the other way, starting by defining the integral of 1/x.

brout

"The derivative is lim (f(x+h)- f(x))/h= axlim {(ah-1)/h}. Notice that that is ax time a limit that is independent of x. That is, as long as the derivative exists, it is ax times a constant. The problem is showing that the lim{(ah-1)/h} EXISTS! And then showing that, if a= 2, that limit is ln(2)."

tell me if i'm wrong, but it doesn't seems so hard to determine this limit..
(a^h-1)/h = (exp (h*ln(a) )-1) / h
= ( 1 + h*ln(a) + o(h*ln(a)) - 1 ) / h h->0
= ln(a) + o(ln(a))
so lim (a^h-1)/h = ln(a) .......

HallsofIvy

Yes, assuming that you know "(exp (h*ln(a) )-1) / h= ( 1 + h*ln(a) + o(h*ln(a)) - 1 ) / h " its easy to do it. Proving what you assumed is the hard part!