real analysis – Uniform convergence of a sequence of functions to a function (g) but their derivatives converge pointwise to a function which is not (g’)

Question is that find a sequence ($f_n$) of continuously differentiable real functions defined on $(0,1)$ converges uniformly to a differentiable function ($g$) and ($f_n’$) converge pointwise to a function that is not ($g’$). I am trying to find this function. I think that $frac{x}{1+x^2n^2}$ converges uniformly to zero on (0,1) and derivate $frac{1-n^2x^2}{(1+x^2n^2)^2}$ converges pointwise to 1 at zero and 0 for all other points. Is my logic correct? Any help will be appreciated.Thanks