Question

This is the technique my Prof used to prove the Power Rule for Derivatives. How would I prove it through this technique? /

Quote (My Prof): I used the factorization of A^n-B^n=(A-B)(A^{n-1}+A^{n-2}B+A^{n-3}B^2+.... +A B^{n-2}+B^{n-1}).

This is the technique my Prof used to prove the Power Rule for Derivatives. How would I prove it through this technique? /

Quote (My Prof): I used the factorization of A^n-B^n=(A-B)(A^{n-1}+A^{n-2}B+A^{n-3}B^2+.... +A B^{n-2}+B^{n-1}).

@Mathematics

Answer 1

\[\frac{d}{dx} (x^n)=\lim_{x \rightarrow x_0}\frac{f(x)-f(x_0)}{x-x_0}=\lim_{x \rightarrow x_0}\frac{x^n-x_0^n}{x-x_0}\]
\[=\lim_{x \rightarrow x_0}\frac{(x-x_0)(x^{n-1}+x^{n-2}x_0+ \cdot \cdot \cdot x x_0^{n-2}+x_0^{n-1})}{x-x_0}\]

Answer 2

What I don't understand is your second line. How does that difference of n's work?

Answer 3

Would you mind explaining as to how you got there with the n's. That is the one part that is confusing to me.

Answer 4

myininaya, what program are you using?