Hi
How do you differentiate using first principles
eg 3^x
Any help will be appreciated
I'd still like to see the response of the OP to post #4. (Many students don't realise that the questions they get given in their textbook etc. are sanitised and cooked to work. A lot of time gets wasetd when it turns out that the posted question has just been plucked out of the air by the student).
I know the process. i just thought it would not be needed.. Here it goes:
$\displaystyle \frac{3^h-1}{h}=\frac{e^z-1}{\frac{z}{\ln 3}}$ (where $\displaystyle z=h\ln 3$)
$\displaystyle =\ln 3\lim_{z\to\0}\frac{e^z-1}{z}=\ln 3$
But now if you say why is $\displaystyle \lim_{z\to\0}\frac{e^z-1}{z}=1 $? , I can't answer because I have learned it as a result without proof.
There are multiple ways that textbook authors present this material. I like the way it is done in Gillman & McDowell.
But theirs is much different from your question, which is the way James Stewart’s books do it.
Stewart simply defines $\displaystyle \mathbf{e}$ to the number such that
$\displaystyle \displaystyle\lim _{h \to 0} \frac{{e^h - 1}}{h} = 1$.