The Total Derivative

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

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