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Structure of quantum corrections in N=1 supersymmetric gauge theories is investigated in the case of using the regularization by higher covariant derivatives. It is demonstrated that this regularization allows revealing some interesting features which lead to the exact relations between the renormalization group functions. In particular, it is proved that the NSVZ equation, which relates the beta-function to the anomalous dimension of the matter superfields, naturally appears in this case. We present the all-loop derivation of this equation and a simple renormalization prescription under which it is valid.