f(x+h)-f(x)

Question

for the function f(x)1/8x the different quotient

Answer 1

If f(x)=⅛x, then f(x+h)=⅛(x+h)=⅛x+⅛h; (f(x+h)-f(x))/h is the difference quotient:

(⅛x+⅛h-⅛x)/h=⅛h/h=⅛, so the difference quotient is ⅛.

If f(x)=1/(8x), then f(x+h)=1/(8x+8h).

(f(x+h)-f(x))/h=[1/(8x+8h)-1/(8x)]/h=

(8x-(8x+8h))/(8xh(8x+8h))=

-8h/(64x²h+64xh²)= (divide top and bottom by 8h)

-1/(8x²+8xh).

As h→0, this becomes -1/(8x²), the derivative of f(x)=1/(8x).