## The Total Derivative

In contrast to The Total Differential, which relates the change in a function
$F(x_1,x_2,...,x_n)$
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{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}