we assume P(k).
so .
now we try to prove P(k+1). remember, we can only use P(k).
, since is just a constant, and we are assuming P(k).
now use your "base case" to finish.
Let n be a natural number. Make a conjecture about the nth derivative of the function f(x) = e^(ax). That is, what is the nth derivative of e^(ax).
Then use mathematical induction to prove your conjecture.
My conjecture is as follows: For every natural number a, the nth derivative of the function f(x) = e^ax is f^n (x) = a^n (e^(ax)).
I have proved the initial step P(1):
f'(x)=ae^(ax) = a^1 e^(ax) so that ae^(ax) = ae^(ax). This proves the initial step. Now here is where I am having trouble proving the inductive step, that is P(k+1) is true. Here is what I have so far
ae^(ax) = a^(n+1) e^(ax)
ae^(ax) = a*a^n*e^ax
I used the fact that a^(n+1) = a*a^n... but I don't know where to go from here, or how to prove the inductive step... any help please?
we assume P(k).
so .
now we try to prove P(k+1). remember, we can only use P(k).
, since is just a constant, and we are assuming P(k).
now use your "base case" to finish.
the "base case" is simply P(1).
P(k+1) would be: , which is what you're trying to prove.
what you wrote is "almost" the P(k+1) case (except you used n instead of k), save for some algebraic manipulation.