Derivatives and quantities with units

The units of quantities are conveniently ignored in the definition of a derivative, but they should not be. A derivative should be defined as a function of two quantities, both with their own units:

\mathbf{x'}(\mathbf{x},t)=\lim_{\Delta t\rightarrow 0}\frac{\mathbf{x}(t+\Delta t)-\mathbf{x}(t)}{\Delta t}

where r’ is a vector function of two quantites and r is a vector functon of one quantity. The second derivative is then a function of three quantities, each with their own units:

\mathbf{x''}(\mathbf{x'},\mathbf{x},t) =\lim_{\Delta t\rightarrow 0}\frac{\mathbf{x'}(\mathbf{x},t+\Delta t)-\mathbf{x'}(\mathbf{x},t)}{\Delta t}

Other derivatives should be defined similarly.