Given f(x)= 4/x. Find f’(x) using the definition of derivative.

Question

recall f’(x)= lim   F(x+h) - f(x) /h

                 h -> o

Answer 1

The numerator is

 

   4/(x+h) - 4/x =

 

 [4x - 4(x+h) ]/ [ x(x+h)]   <--- common denominators

 

[ 4x - 4x - 4h ]/ [ x(x+h)]

 

 [-4h]/[x(x+h)]

 

 

This is OVER denominator h, so the h cancels out:

 

 -4/ [ x(x+h)]

 

Taking the limit as h tends to zero,

 

 -4/[x(x+0)] = -4/[x*x] = -4/x^2 = -4x^(-2)

 

 

This agrees with the power rule:

  Dx [ 4/x] = Dx[ 4x^(-1) ] = -4x^(-2)