Is there a chain rule for partial derivatives?

Question

Scarlett Allison · Accepted Answer

Yes, there is. Suppose you have three functions: #y(x,z)#, #x(t)#, and #z(t)#.  #y = x + z#
#x = 2t#
#z = t^2# The function #y# varies according to #x# and #z#, while #x# and #z# are function themselves that vary with #t#. When you take the partial derivative with respect to #x#, for instance: #(dely)/(delx) = 1*color(green)((delx)/(delx)) + 1*color(green)((delz)/(delx))# But since #z# is not a function of #x#, nothing happens to it. #= 1 + z# So you still see the chain rule, but it may not be obvious, and here it looks a bit redundant. #(delx)/(delx)# merely becomes a #1# multiplier in the equation.  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 #t#, which is like a variable "indirectly associated" with #y# or "embedded" within #y#, it is easier to see the effects of the chain rule here, where you have embedded partial derivatives (inexact differentials) due to the fact you have multiple variables: #(dy)/(dt) = d/(dt)[x + z]# #= (dely)/(delx)cdotcolor(green)((delx)/(delt)) + (dely)/(delz)cdotcolor(green)((delz)/(delt))# 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 #(dely)/(delx) = 1# and #(dely)/(delz) = 1#: #= 1*color(green)((delx)/(delt)) + 1*color(green)((delz)/(delt))# #= color(blue)(2 + 2t)#

Charles Anctil · Answer

undefined Yes, there is a chain rule for partial derivatives, often referred to as the multivariable chain rule. It states that if a function z = f(x, y) depends on variables x and y, where x and y are themselves functions of other variables u and v respectively, then the partial derivative of z with respect to u can be expressed as the sum of the partial derivatives of z with respect to x and y, multiplied by the corresponding partial derivatives of x and y with respect to u and v, respectively. Mathematically, it can be written as ∂z/∂u = (∂z/∂x)(∂x/∂u) + (∂z/∂y)(∂y/∂u).