Problems: The Chain Rule with Constraints | Massachusetts Institute of Technology - Edubirdie

University:
Massachusetts Institute of Technology
Course:
18.02SC | Multivariable Calculus
Academic year:

2019
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312

Pages:

1
Author:

Advinogligo06

Problems: The Chain Rule with Constraints Suppose w = u3 − uv 2 , u = xy and v = u + x. ∂w ∂w 1. Find and using the chain rule. ∂u x ∂x u Answer: In ﬁnding ∂w ∂u x we assume v and y are functions of u and that x is a constant. ∂w ∂u = (3u2 − v 2 ) − u · 2v x 2 ∂v ∂u x 2 = 3u − v − 2uv. Similarly, ∂w ∂x = 0 − u · 2v u ∂v ∂x u = −2uv. 2. Find ∂w ∂u and x ∂w ∂x using diﬀerentials. u Answer: We can compute: dw = (3u2 − v 2 )du − 2uv dv; du = x dy + y dx; dv = du + dx. We’re interested in the independent variables u and x so we substitute dv = du + dx to get: dw = (3u2 − v 2 )du − 2uv(du + dx) = (3u2 − v 2 − 2uv)du − 2uv dx. Using the fact that dw = ∂w ∂u du + x ∂w ∂x dx, we get the expected answer. u Note that we did not need the variable y or the equation u = xy in these calculations!

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