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We introduce a method for construction of a covariant differential calculus over a Hopf algebra $A$ from a quantized calculus $da=[D,a]$, $a\in A$, where $D$ is a candidate for a Dirac operator for $A$. We recover the method of construction of a bicovariant differential calculus given by T. Brzeziński and S. Majid created from a central element of the dual Hopf algebra $A^{\circ }$. We apply this method to the Dirac operator for the quantum $\mathrm{SL}\left(2\right)$ given by S. Majid. We find that the differential calculus obtained by our method is the...