Understanding the Chain Rule Formula

University
Massachusetts Institute of Technology
Course
Multivariable Calculus
Pages

2

Academic year

2023
Author

Sporkz
Views

94

Understanding the Chain Rule Formula So, given two intermediate variables and two final variables that you express thesein terms of, we can write a partial derivative as follows: partial f over partial u equalspartial f over partial x, times partial x over partial u plus partial f over partial y, timespartial y over partial u. The other one is similar but with v instead of u. Partial f,partial x times partial x, partial v, plus partial f, partial y, partial y, partial v. In order to understand these formulas, we need to understand a couple of things.First, how does f depend on u? Well, it depends on x and y. So we'll put partial f,partial x and partial f, partial y. Now x and y are here because they actually dependon u as well. How does x depend on u? The answer is partial x, partial u. How doesy depend on u? The answer is partial y, partial u. So, the structure of these formulasis simple. To find the partial of f with respect to some new variable, you use thepartials with respect to x and y, which in turn depend on v. You multiply them by theirrespective partial derivatives with respect to v, add them together, and that's how youget it. We can describe this relationship using the equation f equals u2 + x/u. If we want tofind out how f changes if we change u a little bit, we can see that it would depend onx and y, which in turn depend on u. If I change u at this rate, how does that cause xto change? Well, the answer is partial x, partial u. And now if I change x at this rate,how does that cause f to change? The rate of change of y is equal to the partial derivative of f with respect to u timesthe partial derivative of y with respect to u. If the value of u changes, then y alsochanges and therefore f also changes. In reality, if you modify either x or y, the otherwill also change. The function f(x,y) describes how x and y affect each other. Theeffect of modifying one or both variables is described by the sum of their individualeffects on f(x,y). But if there are more variables involved, then you can just add moreterms to your sum. Here's another thing that may be a little bit confusing. So what is tempting? Well,what is tempting here would be to simplify these formulas by removing these partialx's. So let's simplify by partial x. Let's simplify by partial y. We get this formula righthere.

The reason the above formula doesn't work is because partial derivatives are not thesame thing as total derivatives. While total derivatives can be simplified, partialderivatives cannot. The curly d in the formula is there to remind us that we must notsimplify anything when dealing with partial derivatives. The following is the simplest formula you can use.