# 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 over the curve , . Write an expression for , giving your answer in terms of the partial derivatives of .

• A
• B
• C
• D
• E

Q2:

The differentiable functions , describe a curve with . Use the chain rule to find an expression for , where .

• A
• B
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• E

Q3:

Let and consider the curve given by . Use the chain rule to determine the values of when . You may keep your answer in terms of .

• A where is an integer
• B where is an integer
• C where is an integer
• D where is an integer
• E where is an integer

Q4:

Let over the curve , , . Write an expression for , giving your answer in terms of the partial derivatives of .

• A
• B
• C
• D
• E

Q5:

The function can be considered in terms of the polar coordinates via the transformation where and . By considering as the second component of , or otherwise, write an expression for this partial derivative in terms of and .

• A
• B
• C
• D
• E