How do you use the limit definition of the derivative to find the derivative of f(x)=x^3?

1 Answer
Steve M
Nov 10, 2016

f'(x) = 3x^2

Explanation:

By definition of the derivative f'(x)=lim_(h rarr 0) ( f(x+h)-f(x) ) / h
So with f(x) = x^3 we have;

f'(x) = lim_(h rarr 0) ( (x+h)^3 - x^3 ) / h
:. f'(x) = lim_(h rarr 0) ( x^3+3x^2h+3xh^2+h^3 - x^3 ) / h
:. f'(x) = lim_(h rarr 0) ( 3x^2h+3xh^2+h^3 ) / h
:. f'(x) = lim_(h rarr 0) ( 3x^2+3xh+h^2 )
:. f'(x) = 3x^2

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