Worksheet: The Chain Rule for Multivariate Functions

In this worksheet, we will practice using the chain rule with partial derivative of multivariable functions, such as finding ∂ f(x,y) / ∂u, where x=g(u,v) and y=h(u,v).

Q1:

Let 𝑧=𝑓(π‘₯,𝑦) over the curve π‘₯=𝑑+1, 𝑦=π‘‘βˆ’1. Write an expression for dd𝑧𝑑, giving your answer in terms of the partial derivatives of 𝑓.

  • A2π‘‘πœ•π‘“πœ•π‘₯(𝑑+1,π‘‘βˆ’1)βˆ’2π‘‘πœ•π‘“πœ•π‘¦(𝑑+1,π‘‘βˆ’1)
  • Bπ‘‘πœ•π‘“πœ•π‘₯(𝑑+1,π‘‘βˆ’1)Γ—π‘‘πœ•π‘“πœ•π‘¦(𝑑+1,π‘‘βˆ’1)
  • C2π‘‘πœ•π‘“πœ•π‘₯(𝑑+1,π‘‘βˆ’1)Γ—2π‘‘πœ•π‘“πœ•π‘¦(𝑑+1,π‘‘βˆ’1)
  • D2π‘‘πœ•π‘“πœ•π‘₯(𝑑+1,π‘‘βˆ’1)+2π‘‘πœ•π‘“πœ•π‘¦(𝑑+1,π‘‘βˆ’1)
  • Eπ‘‘πœ•π‘“πœ•π‘₯(𝑑+1,π‘‘βˆ’1)+π‘‘πœ•π‘“πœ•π‘¦(𝑑+1,π‘‘βˆ’1)

Q2:

The differentiable functions π‘₯=πœ™(𝑑), 𝑦=πœ“(𝑑) describe a curve with (πœ™(1),πœ“(1))=(2,3). Use the chain rule to find an expression for dd𝑀𝑑|||, where 𝑀=𝑒(𝑦)cos.

  • A𝑒(3)πœ“β€²(1)βˆ’π‘’(3)πœ™β€²(1)cossin
  • Bπ‘’πœ™β€²(1)βˆ’(3)πœ“β€²(1)sin
  • C𝑒(3)πœ™β€²(1)βˆ’π‘’(3)πœ“β€²(1)cossin
  • D𝑒(3)πœ™β€²(1)+𝑒(3)πœ“β€²(1)cossin
  • E𝑒(2)πœ™β€²(1)βˆ’π‘’(2)πœ“β€²(1)sincos

Q3:

Let 𝑓(π‘₯,𝑦)=π‘₯+π‘¦οŠ¨οŠ¨ and consider the curve given by πœ™(𝑑)=(2𝑑,𝑑)sincos. Use the chain rule to determine the values of 𝑑 when dd𝑑𝑓(πœ™(𝑑))=0. You may keep your answer in terms of cos.

  • Aπ‘›πœ‹4,π‘›πœ‹βˆ’ο€Ώο„ž12,π‘›πœ‹+ο€Ώο„ž12coscos where 𝑛 is an integer
  • Bπ‘›πœ‹2,π‘›πœ‹βˆ’ο€Ό14,π‘›πœ‹+ο€Ό14coscos where 𝑛 is an integer
  • Cπ‘›πœ‹2,π‘›πœ‹βˆ’ο€Ό25,π‘›πœ‹+ο€Ό25coscos where 𝑛 is an integer
  • Dπ‘›πœ‹4,π‘›πœ‹βˆ’ο€Ό35,π‘›πœ‹+ο€Ό35coscos where 𝑛 is an integer
  • Eπ‘›πœ‹2,π‘›πœ‹βˆ’ο€Ό12,π‘›πœ‹+ο€Ό12coscos where 𝑛 is an integer

Q4:

Let 𝑀=𝑓(π‘₯,𝑦,𝑧) over the curve π‘₯=βˆšπ‘‘, 𝑦=1𝑑, 𝑧=3𝑑. Write an expression for dd𝑀𝑑|||οŠͺ, giving your answer in terms of the partial derivatives of 𝑓.

  • Add𝑀𝑑=14πœ•π‘“πœ•π‘₯ο€Ό2,14,12+116πœ•π‘“πœ•π‘¦ο€Ό2,14,12+3πœ•π‘“πœ•π‘§ο€Ό2,14,12
  • Bdd𝑀𝑑=14πœ•π‘“πœ•π‘₯ο€Ό2,14,12οˆβˆ’116πœ•π‘“πœ•π‘¦ο€Ό2,14,12+3πœ•π‘“πœ•π‘§ο€Ό2,14,12
  • Cdd𝑀𝑑=2πœ•π‘“πœ•π‘₯ο€Ό2,14,12+14πœ•π‘“πœ•π‘¦ο€Ό2,14,12+12πœ•π‘“πœ•π‘§ο€Ό2,14,12
  • Ddd𝑀𝑑=163πœ•π‘“πœ•π‘₯ο€Ό2,14,12+1.386πœ•π‘“πœ•π‘¦ο€Ό2,14,12+24πœ•π‘“πœ•π‘§ο€Ό2,14,12
  • Edd𝑀𝑑=14πœ•π‘“πœ•π‘₯ο€Ό2,14,12οˆβˆ’πœ•π‘“πœ•π‘¦ο€Ό2,14,12+3πœ•π‘“πœ•π‘§ο€Ό2,14,12

Q5:

The function 𝐹(π‘₯,𝑦)=π‘₯+π‘¦οŠ©οŠ¨ can be considered in terms of the polar coordinates via the transformation 𝑐(π‘Ÿ,πœƒ)=(π‘₯,𝑦) where π‘₯=π‘Ÿπœƒcos and 𝑦=π‘Ÿπœƒsin. By considering πœ•πΉπœ•πœƒ as the second component of βˆ‡πΉ(𝑐), or otherwise, write an expression for this partial derivative in terms of π‘₯ and 𝑦.

  • Aπœ•πΉπœ•πœƒ=2π‘₯𝑦+3π‘₯π‘¦οŠ¨
  • Bπœ•πΉπœ•πœƒ=π‘₯𝑦+π‘₯π‘¦οŠ¨
  • Cπœ•πΉπœ•πœƒ=π‘₯𝑦+π‘₯π‘¦οŠ¨οŠ©
  • Dπœ•πΉπœ•πœƒ=2π‘₯π‘¦βˆ’3π‘₯π‘¦οŠ¨
  • Eπœ•πΉπœ•πœƒ=3π‘₯𝑦+π‘₯π‘¦οŠ¨

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