Chain rule Now we will formulate the chain rule when there is more than one independent variable. We suppose w is a function of x, y and that x, y are functions of u, v. That is, w = f (x, y) and x = x(u, v), y = y(u, v). The use of the term chain comes because to compute w we need to do a chain of computa tions (u, v) → (x, y) → w. We will say w is a dependent variable, are intermediate variables. u and v are independent variables and x and y ∂w ∂w and . ∂x ∂y Since, ultimately, w is a function of u and v we can also compute the partial derivatives ∂w ∂w and . The chain rule relates these derivatives by the following formulas. ∂u ∂v Since w is a function of x and y it has partial derivatives ∂w ∂u = ∂w ∂x ∂w ∂y + ∂x ∂u ∂y ∂u ∂w ∂v = ∂w ∂x ∂w ∂y + . ∂x ∂v ∂y ∂v Example: Given w = x2 y + y 2 + x, x = u2 v, y = uv 2 ﬁnd ∂w . ∂u Answer: First we compute ∂w ∂w ∂x ∂y ∂x ∂y = 2xy + 1, = x2 + 2y, = 2uv, = v2 , = u2 , = 2uv. ∂x ∂y ∂u ∂u ∂v ∂v The chain rule then implies ∂w ∂u = ∂w ∂x ∂w ∂y + ∂x ∂u ∂y ∂u = (2xy + 1)2uv + (x2 + 2y)v 2 ∂w ∂v = ∂w ∂x ∂w ∂y + ∂x ∂v ∂y ∂v = (2xy + 1)u2 + (x2 + 2y)2uv. Often, it is okay to leave the variables mixed together. If, for example, you wanted to compute ∂w ∂u when (u, v) = (1, 2) all you have to do is compute x and y and use these ∂w values, along with u, v, in the formula for . ∂u x = 2, y = 4 ⇒ ∂w = (5)(4) + (12)(4) = 68. ∂u If you actually need the derivatives expressed in just the variables u and v then you would have to substitute for x, y and z. Proof of the chain rule: Just as before our argument starts with the tangent approximation at the point (x0 , y0 ). � � ∂w �� ∂w �� Δw ≈ Δx + Δy. ∂x �o ∂y �o Now hold v constant and divide by Δu to get � � Δw ∂w �� Δx ∂w �� Δy ≈ + . Δu ∂x �o Δu ∂y �o Δu Finally, letting Δu → 0 gives the chain rule for ∂w . ∂u Ambiguous notation Often you have to ﬁgure out the dependent and independent variables from context. Thermodynamics is a big player here. It has, for example, the variables P , T , V , U , S. and any two can be taken to be independent and the others are functions of those two. We will do more with this topic in the future.