Posted by Joel on November 13, 2002 at 01:46:04:
In Reply to: Proof by Induction posted by Pete on November 12, 2002 at 17:59:46:
: Problem is show
: d x^n/dx=nx^n-1
: Prove using the defintion dx/dx=1, the product rule and induction.
: I'm having a dumb blonde day and I just can't do it!
: ???
I will use f'(x^n) instead of d x^n/dx for readability.
You want to prove the proposition P(n) that f'(x^n) = n*x^(n-1) given that f'(x) = 1.
Basis step, show that P(1) is true:
f'(x^1) = 1*x^(1-1)
1 = 1*1^0 = 1 so the base case is true.
Inductive step:
Assuming that P(n) is true, show that P(n+1) is true
Assuming: f'(x^n) = n*x^(n-1)
Show that f'(x^(n+1)) = (n+1)*x^(n+1-1) = (n+1)*x^n
f'(x^(n+1)) = f'(x*x^n) [because x^(n+1) = x*x^n]
___________ = (x^n)*f'(x) + x*f'(x^n) [by the product rule]
___________ = x^n + x*f'(x^n) [because f'(x)=1]
___________ = x^n + x*(n*x^(n-1)) [by the inductive hypothesis]
___________ = x^n + n*x^n
___________ = (n+1)* x^n
This shows that if P(n) is true, then P(n+1) is true, completing the proof.