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
\[\begin{equation} \begin{aligned} \frac{dF}{ds} &= \frac{\partial F}{\partial x} \frac{dx}{ds}+ \frac{\partial F}{\partial x} \frac{dx}{ds} \\ &= (2xy)(2s+1)+ (x^2) (\frac{1}{2 \sqrt{s}})\\ &= (2(s^2+s)( \sqrt{s})(2s+1)+(s^2+s)^2 (\frac{1}{2 \sqrt{s}}) \end{aligned} \end{equation}\]

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