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{\frac3{x+h}-\frac3x}h`

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

`\displaystyle f'(x) = \lim_(h\to0)(3x-3(x+h))/(hx(x+h))`

`\displaystyle f'(x) = \lim_(h\to0)(3x-3x-3h)/(hx(x+h))`

`\displaystyle f'(x) = \lim_(h\to0)(-3h)/(hx(x+h))`

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