\[F(x_1,x_2,...,x_n)\]
to incremental changes in each argument \[x_1, \; x_2,..., \; x_n\]
the total derivative relates the rate of change of \[F\]
with respect to each argument to the rate of change of each parameter with respect to some parameter in which it is defined.Suppose
\[F(x,y)=x^2 y\]
then and each of \[x, \; y\]
is defined in terms of a parameter \[s\]
, \[x=s^2+s, \; y= \sqrt{s}\]
then the total derivative of \[F\]
is\[\begin{equation} \begin{aligned} \frac{dF}{ds} &= \frac{\partial F}{\partial x} \frac{dx}{ds}+ \frac{\partial F}{\partial x} \frac{dx}{ds} \\ &= (2xy)(2s+1)+ (x^2) (\frac{1}{2 \sqrt{s}})\\ &= (2(s^2+s)( \sqrt{s})(2s+1)+(s^2+s)^2 (\frac{1}{2 \sqrt{s}}) \end{aligned} \end{equation}\]