Differentiate from first principles

[auru]

Homework Statement

Differentiate from first principles with respect to x: x^n (where n belongs to the natural numbers).

Homework Equations

f'(x) = Lim x→0 [f(x+h) - f(x)]/h

The Attempt at a Solution

f'(x) = Lim x→0 [f(x+h) - f(x)]/h
= Lim x→0 [(x+h)^n - x^n]/h

I need some help simplifying this.

 

[UltrafastPED]

[(x+h)^n - x^n] ... use the binomial expansion theorem to get the first few terms of (x+h)^n:

x^n + n x^(n-1) h^1 + (n|2) x^(n-2) h^2 ... where (n|2) is the number of combinations of n items taken 2 at a time.

Then simplify, and take the limit.

 

[auru]

[(x+h)^n - x^n] ... use the binomial expansion theorem to get the first few terms of (x+h)^n:

x^n + n x^(n-1) h^1 + (n|2) x^(n-2) h^2 ... where (n|2) is the number of combinations of n items taken 2 at a time.

Then simplify, and take the limit.

I don't have a very good understanding of the binomial theorem. I'm not sure what this means: "where (n|2) is the number of combinations of n items taken 2 at a time" or how it helps me.

My only experience of the binomial theorem has been making the h into a 1 thus giving me (x+h)^n = [(1/h)^n].(x/h + 1)^n where n has always been given to me.

 

[UltrafastPED]

Read the Wikipedia article ... http://en.wikipedia.org/wiki/Binomial_theorem

Pascal's Triangle gives you the coefficients. Look at the pattern.

Then divide thru by h and see what is left as h-> 0.

 

[auru]

I tried the above and just came out with x^n-1. I'm not sure where to obtain the n I need.

I'm having some serious issues with [sin(x+h) - sin(x)]/h and {[1/(x+h^1/2)]-[1/(x^1/2)]}. I think the second of which can be solved with (a-b)(a+b) = a^2 - b^2 or am I completely wrong?

 

[Mark44]

I tried the above and just came out with x^n-1. I'm not sure where to obtain the 1/n I need.

Show us what you did when you expanded (x + h)ⁿ using the binomial theorem.

 

[auru]

Show us what you did when you expanded (x + h)ⁿ using the binomial theorem.

I've just managed to do it. I expanded (x+h)^n, subtracted x^n and divided by h. Substituting 0 in for h I am left with just one term which had no h after the division, which was (n 1)x^n-1 which I now realize gives me n.x^n-1.

But I am still unsure about the others I mentioned above. I tried just about every identity I could to get the sine one to work, unless I missed something. I have lim θ→0 Sinθ/θ = 1 but I'm not sure how to make it helpful.