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).


Let 𝑧=𝑓(𝑥,𝑦) over the curve 𝑥=𝑡+1, 𝑦=𝑡1. Write an expression for dd𝑧𝑡, giving your answer in terms of the partial derivatives of 𝑓.

  • A2𝑡𝜕𝑓𝜕𝑥(𝑡+1,𝑡1)2𝑡𝜕𝑓𝜕𝑦(𝑡+1,𝑡1)
  • B𝑡𝜕𝑓𝜕𝑥(𝑡+1,𝑡1)×𝑡𝜕𝑓𝜕𝑦(𝑡+1,𝑡1)
  • C2𝑡𝜕𝑓𝜕𝑥(𝑡+1,𝑡1)×2𝑡𝜕𝑓𝜕𝑦(𝑡+1,𝑡1)
  • D2𝑡𝜕𝑓𝜕𝑥(𝑡+1,𝑡1)+2𝑡𝜕𝑓𝜕𝑦(𝑡+1,𝑡1)
  • E𝑡𝜕𝑓𝜕𝑥(𝑡+1,𝑡1)+𝑡𝜕𝑓𝜕𝑦(𝑡+1,𝑡1)


The differentiable functions 𝑥=𝜙(𝑡), 𝑦=𝜓(𝑡) describe a curve with (𝜙(1),𝜓(1))=(2,3). Use the chain rule to find an expression for dd𝑤𝑡|||, where 𝑤=𝑒(𝑦)cos.

  • A𝑒(3)𝜓(1)𝑒(3)𝜙(1)cossin
  • B𝑒𝜙(1)(3)𝜓(1)sin
  • C𝑒(3)𝜙(1)𝑒(3)𝜓(1)cossin
  • D𝑒(3)𝜙(1)+𝑒(3)𝜓(1)cossin
  • E𝑒(2)𝜙(1)𝑒(2)𝜓(1)sincos


Let 𝑓(𝑥,𝑦)=𝑥+𝑦 and consider the curve given by 𝜙(𝑡)=(2𝑡,𝑡)sincos. Use the chain rule to determine the values of 𝑡 when dd𝑡𝑓(𝜙(𝑡))=0. You may keep your answer in terms of cos.

  • A𝑛𝜋4,𝑛𝜋12,𝑛𝜋+12coscos where 𝑛 is an integer
  • B𝑛𝜋2,𝑛𝜋14,𝑛𝜋+14coscos where 𝑛 is an integer
  • C𝑛𝜋2,𝑛𝜋25,𝑛𝜋+25coscos where 𝑛 is an integer
  • D𝑛𝜋4,𝑛𝜋35,𝑛𝜋+35coscos where 𝑛 is an integer
  • E𝑛𝜋2,𝑛𝜋12,𝑛𝜋+12coscos where 𝑛 is an integer


Let 𝑤=𝑓(𝑥,𝑦,𝑧) over the curve 𝑥=𝑡, 𝑦=1𝑡, 𝑧=3𝑡. Write an expression for dd𝑤𝑡|||, giving your answer in terms of the partial derivatives of 𝑓.

  • Add𝑤𝑡=14𝜕𝑓𝜕𝑥2,14,12+116𝜕𝑓𝜕𝑦2,14,12+3𝜕𝑓𝜕𝑧2,14,12
  • Bdd𝑤𝑡=14𝜕𝑓𝜕𝑥2,14,12116𝜕𝑓𝜕𝑦2,14,12+3𝜕𝑓𝜕𝑧2,14,12
  • Cdd𝑤𝑡=2𝜕𝑓𝜕𝑥2,14,12+14𝜕𝑓𝜕𝑦2,14,12+12𝜕𝑓𝜕𝑧2,14,12
  • Ddd𝑤𝑡=163𝜕𝑓𝜕𝑥2,14,12+1.386𝜕𝑓𝜕𝑦2,14,12+24𝜕𝑓𝜕𝑧2,14,12
  • Edd𝑤𝑡=14𝜕𝑓𝜕𝑥2,14,12𝜕𝑓𝜕𝑦2,14,12+3𝜕𝑓𝜕𝑧2,14,12


The function 𝐹(𝑥,𝑦)=𝑥+𝑦 can be considered in terms of the polar coordinates via the transformation 𝑐(𝑟,𝜃)=(𝑥,𝑦) where 𝑥=𝑟𝜃cos and 𝑦=𝑟𝜃sin. By considering 𝜕𝐹𝜕𝜃 as the second component of 𝐹(𝑐), or otherwise, write an expression for this partial derivative in terms of 𝑥 and 𝑦.

  • A𝜕𝐹𝜕𝜃=2𝑥𝑦+3𝑥𝑦
  • B𝜕𝐹𝜕𝜃=𝑥𝑦+𝑥𝑦
  • C𝜕𝐹𝜕𝜃=𝑥𝑦+𝑥𝑦
  • D𝜕𝐹𝜕𝜃=2𝑥𝑦3𝑥𝑦
  • E𝜕𝐹𝜕𝜃=3𝑥𝑦+𝑥𝑦

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