Optimal recovery of the derivative of the function from its inaccurately given other orders of derivatives and the function itself

The paper deals with the problem of simultaneous recovery of the $k_1$-th and $k_2$-th order derivatives of a function in the mean square norm from inaccurately given derivatives of $n_1$-th and $n_2$-th order and the function itself. The solution is given under some conditions on the errors of given derivatives and the function itself. The problem is solved completely for the case $k_1=k$, $n_1=2k$, $k_2=3k$, $n_2=4k$, $k\in\mathbb N$. It turns out that in contrast to previously encountered situations in the general case, the error of recovery depends on errors of all three errors of input data.