differentiation proof by induction

big_moe

in the leaving cert paper 1, question 6 (c) pdf here

the question was
"prove by induction that d(x^n)/dx = nx^n-1, n>or=1, n E N"

i got as far as "assume true for n=k" but i didnt know how to link the "n=k+1" back with the last line of "n=k". i cant find the solution anywhere, last year www.leavingcertsolutions.com had the solutions online but no signs of them as of yet. i was gonna put this in the leaving cert forum but i thought i would get a better response from in here!

any ideas??
cheers
moe

Crumbs

I think that's a standard proof in the leaving cert course so it should be in your book or notes somewhere.

Basically, to differentiate x^(k+1), change it into x.x^k and use the product rule on that. Simplify and you should end up with what you're looking for, (k+1).x^k

JoseJones

I've uploaded it here:

big_moe

cheers lads! it was only the last 2 lines i had trouble with!

mrskinner

P(k); d/dx x^k = k.x^(k-1) assume to be true.

d/dx(x.x^k) = x.kx^(k-1) + 1.x^k

= k.x^k + x^k

= (k + 1).x^k ............This implies that P(k+1) is true

as d/dx(x^k+1) = (k + 1).x^k

Thus P(n) is true for n.......


ok?

MathsManiac

And it may also interest you to know that solutions are included in the marking schemes, which are available to dowload from the State Examinations Commission's website: www.examinations.ie

(Click on "Exam material archive" and select "marking schemes")