prove derivative of x^n

Question

I need help proving this derivative using the difference quotient.

 

 

prove derivative of x^n

Answer 1

Method 1.

(x^n)'

= lim{y-->x}(y^n - x^n)/(y-x)

= lim{y-->x}[y^(n-1)+y^(n-2)x + ... + x^(n-1)], using long division

= nx^(n-1)

 

Method 2.

(x^n)'

= lim{h-->0}[(x+h)^n - x^n)/h

= lim{h-->0}[(x^n+nx^(n-1)h+O(h) - x^n)/h, binomial expansion

= nx^(n-1), since lim{h-->0}O(h)/h = 0.

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Attn: O(h) means higher degree of zeros than h as h-->0.