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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
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