Multivariable Chain Rule. Dividing Differentials

University
Massachusetts Institute of Technology
Course
Multivariable Calculus
Pages

1

Academic year

2022
Author

Sporkz
Views

74

p {margin: 0; padding: 0;} .ft00{font-size:20px;font-family:Arial;color:#000000;} .ft01{font-size:18px;font-family:ArialMT;color:#000000;} .ft02{font-size:14px;font-family:CambriaMath;color:#000000;} .ft03{font-size:20px;font-family:CambriaMath;color:#000000;} .ft04{font-size:18px;line-height:23px;font-family:ArialMT;color:#000000;} .ft05{font-size:18px;line-height:22px;font-family:ArialMT;color:#000000;} Multivariable Chain Rule. Dividing Differentials The notation suggests that we can also divide everything by some variable thateverybody depends on. For example, if x, y, and z actually depend on someparameter t, then they will vary at a certain rate—dx dt, dy dt, dz dt. By dividingeverything by dt in here, we get df dt.The third thing we can do is divide by something like dt, giving us an infinitesimalrate of change. So df dt equals f sub x dx dt plus f sub y dy dt plus f sub z dz dt. 𝑑𝑓 𝑑𝑡 = 𝑓 𝑥 𝑑𝑥 𝑑𝑡 + 𝑓 𝑦 𝑑𝑦 𝑑𝑡 + 𝑓 𝑧 𝑑𝑧 𝑑𝑡 When t corresponds to x, y, and z, the situation is that a function of t equals each oftheir respective functions. 𝑥 = 𝑥 𝑡 ( ), 𝑦 = 𝑦 𝑡 ( ), 𝑧 = 𝑧(𝑡) If you plug in these values into f, then you can find the value of f. The rate of changewith t of a value of f depends on t. The chain rule is one instance of a general rule that relates the derivatives of adependent variable to the rate-of-change of the variable and its relationship withother dependent variables. It's one instance of a chain rule, which tells you when youhave a function that depends on something and that something in turn depends onsomething else how to find the rate of change of a function on the new variable interms of the derivatives of the function and also the dependence between thevarious variables.