It can be a silly question but I could not be sure at all.

I have a function as follows ;


I know that $z = F(c)$. So, in $z$, there is some term $c$.

When I take the partial derivative of $f(c,z)$, I think I must take into account the variation of $z$ which depends on $c$. Am I right ?

Because when I think to the logic of partial derivative, we just consider the variation of the variable of interest (which is $c$ in this case.) and the other terms are held constant.

Thanks in advance.


It depends "when" in the problem you are taking the derivative.

If you are given a function $f(c, z)$ and told to take $\frac{\partial f}{\partial c}(c, z)$, you must ignore $z$. If you are given a function $g(c) = f(c, F(c))$ and want to find $\frac {\partial g} {\partial c}(c)$, then you need to take into account the original $c$ as well as the $c$'s from the $z$ terms.

  • $\begingroup$ thanks for the clear answer. @anorton $\endgroup$ – optimal control Dec 15 '14 at 13:11

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