How do you use the limit definition of the derivative to find the derivative of $f (x) = 3 x^{2}$ $f(x)=3x^2$ ?

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Answer 1 · 2016-12-24T15:11:18

Use $f'(x) = lim_(h-> 0) (f(x+ h) - f(x))/h$ .

$f'(x) = lim_(h-> 0) (3(x + h)^2 - 3x^2)/h$

$f'(x) = lim_(h-> 0) (3(x^2 + 2xh + h^2) - 3x^2)/h$

$f'(x) = lim_(h-> 0) (3x^2 + 6xh + 3h^2 - 3x^2)/h$

$f'(x) = lim_(h-> 0) (6xh + 3h^2)/h$

$f'(x) = lim_(h-> 0) (h(6x+ 3h))/h$

$f'(x)= lim_(h-> 0) 6x + 3h$

$f'(x) = 6x + 3(0)$

$f'(x) = 6x$

Verification using the power rule yields the same result.

Hopefully this helps!

How do you use the limit definition of the derivative to find the derivative of f(x)=3x^2f(x)=3x2f(x)=3x^2?