3.8. The chain rule

The chain rule also has a finite differences version. That version is

$$\begin{split} \left( \operatorname{Dg} (h(\boldsymbol{q})) \right)\_{j} &= \frac{\operatorname{g} \Big( h \Big( \boldsymbol{q}\_{j+1} \Big) \Big) - \operatorname{g} \Big( h \Big( \boldsymbol{q}\_{j-1} \Big) \Big)}{2 \chi \langle \boldsymbol{v}, \Delta \rangle} \\ &= \frac{\operatorname{g} \Big( h \Big( \boldsymbol{q}\_{j+1} \Big) \Big) - \operatorname{g} \Big( h \Big( \boldsymbol{q}\_{j-1} \Big) \Big)}{2 \chi \Big( \boldsymbol{v}, h \Big( \boldsymbol{q}\_{j+1} \Big) - h \Big( \boldsymbol{q}\_{j} \Big) \Big)} \frac{2 \chi \left( \boldsymbol{v}, h \Big( \boldsymbol{q}\_{j+1} \Big) - h \Big( \boldsymbol{q}\_{j} \Big) \Big)}{2 \chi \langle \boldsymbol{v}, \Delta \rangle} \\ &= (D \boldsymbol{g}(h))\_{j} \frac{\chi \Big( \boldsymbol{v}, h \Big( \boldsymbol{q}\_{j+1} \Big) - h \Big( \boldsymbol{q}\_{j} \Big) \Big)}{\chi \langle \boldsymbol{v}, \Delta \rangle} \end{split} \tag{48}$$

where

Matrices Which are Discrete Versions of Linear Operations http://dx.doi.org/10.5772/intechopen.74356 31

$$(Dg(h))\_{\dot{\jmath}} \coloneqq \frac{g\left(h\left(q\_{\dot{\jmath}+1}\right)\right) - g\left(h\left(q\_{\dot{\jmath}-1}\right)\right)}{2\chi\left(v, h\left(q\_{\dot{\jmath}+1}\right) - h\left(q\_{\dot{\jmath}}\right)\right)}\tag{49}$$

is a finite differences derivative of g hð Þ with respect to h, and the second factor approaches the derivative of h qð Þ with respect to q

$$\frac{\chi\left(\upsilon, h\left(q\_{j+1}\right)-h\left(q\_{j}\right)\right)}{\chi\left(\upsilon, \Delta\right)} \approx \frac{h\left(q\_{j+1}\right)-h\left(q\_{j}\right)+O\left(\Delta h^{2}\right)}{\Delta + O\left(\Delta^{2}\right)}.\tag{50}$$

Thus, we will recover the usual chain rule for continuous variable functions in the limit Δ ! 0.
