How do you find the derivative of f(x)=4x^2f(x)=4x2 using the limit definition?

1 Answer
Steve M
Oct 28, 2016

f'(x)= 8x

Explanation:

By definition, f'(x)= lim_(h->0)(f(x+h)-f(x))/h

So for f(x)=4x^2 we have

f'(x)= lim_(h->0)({4(x+h)^2 - 4x^2})/h
:. f'(x)= lim_(h->0)({4(x^2+2hx+h^2) - 4x^2})/h
:. f'(x)= lim_(h->0)({4x^2+8hx+4h^2 - 4x^2})/h
:. f'(x)= lim_(h->0)(8hx+4h^2 )/h
:. f'(x)= lim_(h->0)(8x+4h )
:. f'(x)= 8x

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