Question

Find derivative of 3x^2+4 using limits

Answer 1

The derivative of a function f(x) is defined as

$f'(x)=\displaystyle\lim_{h\to0}\frac{f(x+h)-f(x)}h$

For f(x) = 3x ² + 4, we have

$f'(x)=\displaystyle\lim_{h\to0}\frac{(3(x+h)^2+4) - (3x^2+4)}h$

$f'(x)=\displaystyle\lim_{h\to0}\frac{(3(x^2+2xh+h^2) - 3x^2}h$

$f'(x)=\displaystyle\lim_{h\to0}\frac{6xh+3h^2}h$

$f'(x)=\displaystyle\lim_{h\to0}(6x+3h) = \boxed{6x}$