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The equivalence between theories depending on the derivatives of $R$, i.e.$f\left( R,\nabla R,...,\nabla^{n}R\right) $, and scalar-multi-tensorialtheories is verified. The analysis is done in both metric and Palatiniformalisms. It is shown that $f\left( R,\nabla R,...,\nabla^{n}R\right) $theories are equivalent to scalar-multi-tensorial ones resembling Brans-Dicketheories with kinetic terms $\omega_{0}=0$ and $\omega_{0}= - \frac{3}{2}$ formetric and Palatini formalisms respectively. This result is analogous to whathappens for $f(R)$ theories. It is worthy emphasizing that thescalar-multi-tensorial theories obtained here differ from Brans-Dicke ones dueto the presence of multiple tensorial fields absent in the last. Furthermore,sufficient conditions are established for $f\left( R,\nablaR,...,\nabla^{n}R\right) $ theories to be written as scalar-multi-tensorialtheories. Finally, some examples are studied and the comparison of $f\left(R,\nabla R,...,\nabla^{n}R\right) $ theories to $f\left( R,\BoxR,...\Box^{n}R\right) $ theories is performed.

physical review. d/physical review. d.

Scalar-multi-tensorial equivalence for higher order $f\left(  R,\nabla_{\mu}  R,\nabla_{\mu_{1}}\nabla_{\mu_{2}}R,...,\nabla_{\mu_{1}}...\nabla_{\mu_{n}  }R\right)$ theories of gravity