Worksheet: Partial Derivatives and the Fundamental Theorem of Calculus

In this worksheet, we will practice finding the partial derivative of a function defined by an integral by applying the fundamental theorem of calculus and Leibniz integral rule.

Q1:

Find the first partial derivative with respect to 𝑥 of the function 𝐹 ( 𝑥 , 𝑦 ) = 𝑒 𝑡 . c o s d

  • A s i n ( 𝑒 )
  • B c o s ( 𝑒 )
  • C c o s ( 𝑒 )
  • D ( 𝑒 ) c o s
  • E ( 𝑒 ) s i n

Q2:

Find the first partial derivative with respect to 𝑦 of the function 𝐹 ( 𝑥 , 𝑦 ) = 𝑒 𝑡 . c o s d

  • A ( 𝑒 ) s i n
  • B c o s ( 𝑒 )
  • C ( 𝑒 ) c o s
  • D c o s ( 𝑒 )
  • E s i n ( 𝑒 )

Q3:

Find the first partial derivative with respect to 𝑦 of the function 𝐹 ( 𝑥 , 𝑦 ) = 𝑡 + 1 𝑡 . d

  • A 𝑦 + 1
  • B 𝑥 + 1
  • C 𝑦 + 1
  • D 𝑥 + 1
  • E 𝑡 + 1

Q4:

Find the first partial derivative with respect to 𝑥 of the function 𝐹 ( 𝑥 , 𝑦 ) = 𝑡 + 1 𝑡 . d

  • A 𝑥 + 1
  • B 𝑥 + 1
  • C 𝑦 + 1
  • D 𝑡 + 1
  • E 𝑦 + 1

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