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On $[0,1]$, the polynomials $p_n(x)=x^n/n$ converge in supremum norm to the zero function, since $\Vert p_n{\Vert }_{\infty }=1/n$. But $p_n^{\prime }(1)=1$ for every $n$, whereas the zero function has left derivative zero at that endpoint. Thus uniform convergence of function values does not imply uniform convergence of derivatives. This does not contradict uniform convergence of Bernstein polynomial derivatives, which has a particular approximation construction and additional differentiability hypotheses.