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Something they never ask you to do in a calculus class...?
1. Using the limit definition of the derivative, prove that the derivative of e^x is in fact e^x
2. Using the limit definition of the derivative, prove that the derivative of ln(x) is in fact 1/x
No, you may not cheat and use L'Hospital's Rule as that involves derivatives.
3. Explain why L'Hospital's rule works
1 Answer
- BhaskarLv 41 decade agoFavorite Answer
f(x) = e^x
f'(x) = lim [ e^(x+h) - e^x ] / h
h-->0
= e^x lim [ e^h - 1] / h.
e^h = 1 + h + h^2/2 + h^3 / 3! + h^4/4! + ...
thus,
f'(x) = e^x lim [ 1+h+(h^2/2)+.... - 1 ] /h
= e^x lim [h + h^2/2 + ...] / h
= e^x lim [1 + h^2/2 + h^3/3! + ...]
h-->0
= e^x* 1
= e^x.
2) f(x) = ln(x).
f'(x) = lim [ ln(x+h) - ln(x) ] / h
h-->0
= lim [ ln (1 + h/x) ] / h
for small values of x,
ln(1 + h/x) = (h/x) - 1/2(h/x)^2 + 1/3(h/x)^3 + ....
substituting this,
f'(x) = lim [(h/x) - 1/2(h/x)^2 + ... ] / h
= lim [ 1/x - h/2x^2 + h^2/3x^3 + ... ]
h-->0
= 1/x.
