Berdasarkan formula lim h->0 (f(x+h)-f(x))/(h) tentukan

Turunan dari f(x)=(x)/(x-5) adalah f'(x) = -5/(x-5)² dan turunan dari f(x)=(x+3)/(x) adalah f'(x) = -3/x².

Pembahasan

Untuk menentukan turunan pertama (f'(x)) dari fungsi yang diberikan menggunakan definisi turunan, yaitu lim h->0 (f(x+h)-f(x))/(h):

a. f(x) = x / (x-5)
   f(x+h) = (x+h) / ((x+h)-5)
   f(x+h) - f(x) = [(x+h) / (x+h-5)] - [x / (x-5)]
   = [(x+h)(x-5) - x(x+h-5)] / [(x+h-5)(x-5)]
   = [x² - 5x + hx - 5h - x² - hx + 5x] / [(x+h-5)(x-5)]
   = -5h / [(x+h-5)(x-5)]
   
   (f(x+h)-f(x))/(h) = [-5h / [(x+h-5)(x-5)]] / h
   = -5 / [(x+h-5)(x-5)]
   
   lim h->0 (f(x+h)-f(x))/(h) = lim h->0 [-5 / [(x+h-5)(x-5)]]
   f'(x) = -5 / [(x-5)(x-5)]
   f'(x) = -5 / (x-5)²

b. f(x) = (x+3) / x
   f(x+h) = ((x+h)+3) / (x+h)
   f(x+h) - f(x) = [(x+h+3) / (x+h)] - [(x+3) / x]
   = [x(x+h+3) - (x+3)(x+h)] / [x(x+h)]
   = [x² + hx + 3x - (x² + hx + 3x + 3h)] / [x(x+h)]
   = [x² + hx + 3x - x² - hx - 3x - 3h] / [x(x+h)]
   = -3h / [x(x+h)]
   
   (f(x+h)-f(x))/(h) = [-3h / [x(x+h)]] / h
   = -3 / [x(x+h)]
   
   lim h->0 (f(x+h)-f(x))/(h) = lim h->0 [-3 / [x(x+h)]]
   f'(x) = -3 / (x*x)
   f'(x) = -3 / x²