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The Chain Rule and Second Partials

The goal of this page is describe the process of finding second partial derivatives with the Chain Rule.

Introduction

To begin, let's find a first partial derivative with the Chain Rule. Suppose we have the function w(x,y) where x and y are functions of s. The Chain Rule helps us find ws. That is, ws = wx xs + w yys. For example, let w = w(x,y), x=sin(r), and y= 2s + r. Let's find wr. Notice, xr = cos(r), yr = 1. So, wr = wx xr + w yyr = wxcos(r) + wy.

Now, let w = w(x,y), x = sr, and y = s2r3. Find wr. Once you have an answer, click here.

Now, let's also recall the product rule. We will use this later. Suppose, h(x,y) = f(x,y)g(x,y). Then hx = fxg + gxf. For example, let h(x,y) = ycos(xy). Then hy = cos(xy) - xysin(xy).

To review this topic, let h(x,y) = ye-xy. Find hy. When you have an answer, click here.

Finding Second Partials

Suppose we wanted to find hyy where h(x,y) = ye-xy, as in the previous example. Then we take hy = e -xy - xye-xy and find its partial derivative with respect to y. Notice, if we let f(x,y) = e-xy - xye-xy, then fy = hyy. Find hyy and click here to check your answer.

Now, we will use the Chain Rule to find a second partial derivative. Suppose w = w(x,y), x = s2 and y = rs. Suppose we want to find wrs. (Note, we could find wsr, as well.) We will look for our answer in terms of partial derivatives taken with respect to x and y.

First, we want to find wr. Recall, wr = wxxr + wyyr. Now xr = 0, and yr = s. So, wr = wys.

Now, we want to find wrs. That is, we want to find the partial derivative of wr with respect to s. Now, wr is the product of two functions. So, we need to use the product rule. Now, the partial derivative of wy with respect to s is wys. The partial derivative of s with respect to s is 1. So, the product rule yields

wrs = wys s + wy.

Now, we want our answer in terms of partial derivatives in terms of x and y. So, the term wys indicates we have more work to do.

Let's use the Chain Rule on wys. That is, wys = w yxxs + wyyy s. Notice, we are finding the partial derivative of wy with respect to s with the Chain Rule. Now, xs = 2s and ys = r. So, w ys = wyx 2s + wyyr.

So, we substitute wys = wyx2s + wyyr into wrs = wyss + wy, and find our solution wrs = 2s2wyx + wyyrs + wy.

Now, to practice these skills, find wsr where w = w(x,y), x = s + 2r, and y = s3 - rs. When you have an answer click here.

Summary

Remember, to find second partials with the Chain Rule you extend your knowledge of finding first partials. With some practice, you will find the Chain Rule a powerful tool in your Calculus bag of skills.

Last modified: March, 1997
Created by Tim Chartier