posted by ibk .
find derivative of f(x)=1/sqx^2+1 by using the definition of derivatives
Do you mean
f(x)= 1/sqrt(x^2+1) = (x^2)+1)^-1/2?
The derivative would be
f'(x) = (-1/2)(2x)/(x^2 +1)^(3/2)
= -x/(x^2 +1)^(3/2)
The "chain rule" was used.
df/dx = df/du*du/dx
with u(x) = u^2 +1 and f(u) = u^-1/2