Example of the Chain Rule

University
Massachusetts Institute of Technology
Course
Multivariable Calculus
Pages

2

Academic year

2023
Author

Nayborsenv
Views

67

Example of the Chain Rule Let us take the function x squared times y plus y as an example. 𝑤 = 𝑥 2 𝑦 + 2 Let's say that maybe x will be t, y will be e to the t, and z will be sine t. 𝑥 = 𝑡, 𝑦 = 𝑒 𝑡 , 𝑧 = 𝑠𝑖𝑛 𝑡 The chain rule states that the derivative of a composite function is equal to thederivative of the function being composed, multiplied by the derivative of each inputvariable. In other words, we take the partial derivative with respect to w sub x timesdx dt plus the partial derivative of w sub z with respect to z dt. Now let's plug in the actual values for these things. So x is t, y is e to the t, so thatwill be 2 t e to the t. dx dt is 1 plus x squared is t squared, and dy dt is e to the t plusdz dt is cosine t. 𝑑𝑤 𝑑𝑡 = 2𝑡𝑒 𝑡 1 + 𝑡 2 𝑒 𝑡 + 𝑐𝑜𝑠 𝑡 So at the end of the calculation, we get 2 t e to the t plus t squared e to t plus cosinet. 𝑑𝑤 𝑑𝑡 = 2𝑡𝑒 𝑡 + 𝑡 2 𝑒 𝑡 + 𝑐𝑜𝑠 𝑡 The chain rule can be used to find the derivative of a composition of functions. Thevalue of w can be determined by plugging in values for t, x and y into thecomposition of functions. In many cases, it is possible to simply plug in values for the variables in a function orequation to verify results. However, if the function or equation is complicated or youdon't know its formula, you may need to use the chain rule. Let's review. The other method of solving this equation is to substitute x = t. So w =x2y + z. 𝑤 = 𝑥 2 𝑦 + 𝑧 The equation for the sine curve is e to the t plus z was sine t.

𝑤 𝑡 ( ) = 𝑡 2 𝑒 𝑡 + 𝑠𝑖𝑛 𝑡 Therefore, we know how to take the derivative of a function with a single variableusing single variable calculus. The derivative of t2 is 2t times et + t2 times e to t. Thederivative of e to t is e to t + cosine of t. 𝑤 𝐼 𝑡 ( ) = 2𝑡𝑒 𝑡 + 𝑡 2 𝑒 𝑡 + 𝑐𝑜𝑠 𝑡 What kind of object is w? In this case, we can think of w as just another variablethat's given as a function of x, y, and z. So you have a function of x, y, z defined bythis formula. Then you can substitute t instead of x, y, z. One can think of w as a function of three variables: t, x, and y. When one plugs in thedependence of these three variables on t, then w becomes a function of t alone.