Question

Find the first principle derivative for 3/x with respect to x​

Answer 1

Given f(x) = 3/x, its derivative is

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

$\displaystyle f'(x) = \lim_{h\to0}\frac{\dfrac3{x+h}-\dfrac3x}h$

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

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

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

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

$\displaystyle f'(x) = \lim_{h\to0}\frac{-3}{x(x+h)} = \boxed{-\frac3{x^2}}$