Using the Binomial Theorem and the dfinition of the derivative of a function

[ryanj123]

Using the Binomial Theorem and the definition of the derivate of a function

f(x) as f'(x)= lim as h tends to 0 ((f(x+h)-f(x))/h)

Prove that if f(x)=x^n

then

f'(x)=nx^(n-1)



I'm confused as to how to exactly incorporate the nCr "n choose r" into this interpretation of the derivative.

Any hints or explanations would be greatly appreciated!

 

[Russell Berty]

Write nCr using its definition:

nCr = n! / [r! * (n-r)!]
Notice that nC1 = n! / [1! * (n-1)!] = (n*(n-1)* ... *2*1)/[(n-1)*(n-2)*...*2*1] = n

Also, the first term in the binomial expansion of (x + h)^n is x^n, so that will disappear when you subtract f(x) = x^n.

The remaining terms all have a factor of h... interesting...