Question

if f(x)= x+(1/x), find f'(x) using the limit definition of derivative.

Please show steps!!!

Thank you!!!

Answer 1

f(x+h) = (x + h) + 1/(x+h)
= [(x+h)^2 + 1] /(x+h)
= [x^2 + 2 x h + h^2 + 1 ] / (x+h)

f(x) = (x^2+1)/x

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

= x^3+2x^2h+h^2x+x -(x^2+1)(x+h)
--------------------------------
x^2 + hx

divide by h
x^3+2x^2h+h^2x+x -(x^2+1)(x+h)
--------------------------------
h x^2 + h^2 x

multiply numerator out
x^3+2x^2h+h^2x+x - x^3 -x -x^2h-h
------------------------------------
h x^2 + h^2 x

x^2 h + h^2 x -h
=----------------
h x^2 + h^2 x

= x^2 + h x -1
---------------
x^2 + h x

let h ---->0

(x^2-1)
---------
x^2

which is actually the right answer believe it or not