Find the derivative of function using the definition of derivative. f(x) =(2 + x)/(1 − 2x).State the domain of the function (interval notation).State the domain of its derivative (interval notation.)

Answer 1

f(x+h) = (2+x+h)/(1-2x-2h)

f(x+h) - f(x) =

(2+x+h)/(1-2x-2h) - (2+x)/(1-2x)

= {(2+x+h)(1-2x) - (2+x)(1-2x-2h)} / [ (1-2x-2h)(1-2x) ]

= { 2+x+h - 4x -2x^2 - 2xh - 2 + 4x + 4h -x + 2x^2 + 2xh}/ [ (1-2x-2h)(1-2x) ]

= (5h)/[ (1-2x-2h)(1-2x) ]

Dividing by h, the difference quotient is:

5/ [ (1-2x-2h)(1-2x) ]

The limit of the difference quotient as h tends to zero is :

5/ (1-2x)^2

Quotient rule says:

[(1-2x) - (-2)(2+x)]/(1-2x)^2 =

[1 - 2x + 4 + 2x]/(1-2x)^2

5/(1-2x)^2

the domain of the original function as well as the derivative is all reals EXCEPT x=1/2