# Is there a chain rule for partial derivatives?

##### 1 Answer

Yes, there is.

Suppose you have three functions:

#y = x + z#

#x = 2t#

#z = t^2#

The function

#(dely)/(delx) = 1*color(green)((delx)/(delx)) + 1*color(green)((delz)/(delx))#

But since

#= 1 + z#

So you still see the chain rule, but it may not be obvious, and here it looks a bit redundant.

*This is similar to the chain rule you see when doing related rates, for instance.*

However, if you take the exact differential with respect to *embedded partial derivatives* (inexact differentials) due to the fact you have multiple variables:

#(dy)/(dt) = d/(dt)[x + z]#

You may see that this is a convenient notation that allows you to see the rationale for formulating the way you take these partial derivatives:

#(dy)/(dt) = (dely)/cancel(delx)cdotcolor(green)(cancel(delx)/(delt)) + (dely)/cancel(delz)cdotcolor(green)(cancel(delz)/(delt))#

That aside, you get, from

#= 1*color(green)((delx)/(delt)) + 1*color(green)((delz)/(delt))#

#= color(blue)(2 + 2t)#