Worksheet: Second- and Higher-Order Partial Derivatives

In this worksheet, we will practice finding the second- and higher-order derivatives of a multivariable function including mixed partial derivatives.

Q1:

Consider the function 𝑓 : given by the formula 𝑓 ( 𝑥 , 𝑦 ) = 𝑥 + 𝑦 ( 𝑥 + 𝑦 + 1 ) . Which of the statements below is incorrect?

  • A 𝑓 is differentiable at the origin.
  • BThe third order partial derivatives satisfy the following equation 𝜕 𝑓 𝜕 𝑥 + 𝜕 𝑓 𝜕 𝑥 𝜕 𝑦 + 𝜕 𝑓 𝜕 𝑥 𝜕 𝑦 + 𝜕 𝑓 𝜕 𝑦 = 0 .
  • C 𝑓 is continuous at the origin.
  • D 𝑓 has a minimum at the origin.

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