derivative of logarithms?

Question

I need the formulas of the derivative of
e^u
log(a)u <--I dont know how to write it, it is meant to be logarithm with base a of the number u (the a is subindex)
a^u
ln u

I have maths exam tomorrow and I think all the formulas I have are wrong because my teacher confuses logarithms, but the exam is not applied by my teacher so I need the correct formulas, please if you have a good maths book somewhere could you share the formulas with me? thnx

Answer 1

derivative of e^u = e^u du
derivative of ln(u) = du/u
a^u=e^{uln(a)}, so the derivative of a^u = e^{uln(a)} ln(a) = ln(a) a^u
s

Answer 2

The derivative of e^u is e^u du.
The derivative of log(a)u is du/(u lna).
The derivative of a^u is a^u lna du.
The derivative of ln |u| = du/u.

As far as I know these are right. I got them from my old calculus book.

Answer 3

d(e^x)/dx=e^x
d(ln x)/dx = 1/x

The other formulas follow from these:

d(a^x)/dx = d(e^(x ln a))/dx = e^(x ln a) * ln a = a^x ln a
d(log_a x)/dx = d(ln x/ln a)/dx = 1/(x ln a)

Answer 4

Assume u is a function of x.
d/dx {e^u} = e^u (du/dx)
d/dx {loga(u)} = (1/u) loga(e) (du/dx)
d/dx {a^u} = a^u ln(a) (du/dx)
d/dx {ln u} = (1/u) (du/dx)