Alternative method for Derivatives

Question

Let f(x) = 5/(x+3)

A) . Reduce with the definition f(x)-f(1)/x-1 so that there is only one x.

B) . Use that answer from a to find derivative f'(1)

Answer 1

Your difference quotient is incorrect

f(1+h) = 5 / (1+h+3) = 5/(4+h)

f(1) = 5/4

f(1+h) - f(1) = 5/(4+h) - 5/4 = [ 5*4 - 5(4+h)]/ [4(4+h)]

= [ 20 - 20 - 5h]/ [ 16 + 4h]

= -5h/( 16 + 4h)

dividing by h:

-5/(16+ 4h)

as h-->0 , the difference quotient tends to -5/16

now , differentiating for real:

y = 5(x+3)^(-1)

y' = -5(x+3)^(-2) = -5/(x+3)^2

f ' (1) = -5/16