Worksheet: Partial Derivatives

In this worksheet, we will practice finding the partial derivatives of multivariable functions.

Q1:

Find the first partial derivative of the function 𝑓 ( 𝑥 ) = 𝑥 + 𝑦 4 with respect to 𝑥 .

  • A 2 𝑥 ( 𝑥 + 𝑦 4 )
  • B 1 3 ( 𝑥 + 𝑦 4 )
  • C 2 𝑥 + 3 ( 𝑥 + 𝑦 4 ) d d
  • D 2 𝑥 3 ( 𝑥 + 𝑦 4 )
  • E 2 𝑥 + ( 𝑥 + 𝑦 4 ) d d

Q2:

Find the first partial derivative with respect to 𝑦 of 𝑓 ( 𝑥 , 𝑦 , 𝑧 ) = 𝑥 𝑦 𝑧 + 2 𝑦 𝑧 .

  • A 3 𝑥 𝑦 𝑧 + 𝑥 𝑧 + 2 𝑥 𝑦 𝑧 + 2 𝑧 + 2 𝑦
  • B 2 𝑥 𝑦 𝑧 + 2 𝑦
  • C 3 𝑥 𝑦 𝑧
  • D 𝑥 𝑧 + 2 𝑧

Q3:

Find the first partial derivative with respect to 𝑥 of the function 𝑓 ( 𝑥 , 𝑦 ) = 𝑥 + 2 𝑦 .

Q4:

Find the first partial derivative with respect to 𝑦 of the function 𝑓 ( 𝑥 , 𝑦 ) = 𝑥 + 2 𝑦 .

Q5:

Find the first partial derivative with respect to 𝑥 of the function 𝑓 ( 𝑥 , 𝑦 ) = 𝑥 + 𝑦 .

  • A 𝑥
  • B 2 𝑦
  • C 2 𝑥 + 2 𝑦
  • D 2 𝑥
  • E 𝑥 + 2 𝑦

Q6:

Find the first partial derivative with respect to 𝑦 of the function 𝑓 ( 𝑥 , 𝑦 ) = 𝑥 𝑦 + 6 𝑥 𝑦 + 4 𝑥 8 𝑦 + 2 .

  • A 2 𝑥 𝑦 𝑥 ( 𝑦 3 𝑥 + 4 ) + 2 𝑦 + 2 d d
  • B 2 ( 𝑥 + 3 𝑦 + 2 )
  • C 2 ( 3 𝑥 𝑦 4 )
  • D 2 ( 4 𝑥 𝑦 2 )
  • E 2 ( 𝑥 + 2 𝑦 2 )

Q7:

Find the first partial derivative with respect to 𝑥 of the function 𝑓 ( 𝑥 , 𝑦 ) = 𝑥 + 5 𝑥 𝑦 .

  • A 4 𝑥 + 5 𝑦
  • B 𝑥 + 1 5 𝑥 𝑦
  • C 4 𝑥 + 5 𝑥 𝑦
  • D 4 𝑥 + 1 5 𝑥 𝑦 + 5 𝑦
  • E 1 5 𝑥 𝑦

Q8:

Find the first partial derivative with respect to 𝑦 of 𝑓 ( 𝑥 , 𝑦 ) = 𝑥 𝑦 3 𝑦 .

  • A 𝑥 + 2 𝑥 𝑦 1 2 𝑦
  • B 2 𝑥 𝑦 3 𝑦
  • C 𝑥 1 2 𝑦
  • D 𝑥 𝑦 1 2 𝑦
  • E 2 𝑥 𝑦

Q9:

Find the first partial derivative with respect to 𝑥 of the function 𝑓 ( 𝑥 , 𝑦 ) = 𝑒 + 𝑥 𝑦 .

  • A 𝑥 ( 𝑒 + 1 )
  • B 𝑥 ( 𝑒 + 1 )
  • C 𝑦 ( 𝑒 + 1 )
  • D 𝑦 ( 𝑒 + 1 )
  • E 𝑥 𝑦 𝑥 + 𝑦 ( 𝑒 + 1 ) d d

Q10:

Find the first partial derivative with respect to 𝑦 of the function 𝑓 ( 𝑥 , 𝑦 ) = 𝑥 .

  • A 𝑥
  • B 4 𝑥
  • C 4 𝑥 𝑦
  • D0
  • E 4 𝑥

Q11:

Find the first partial derivative with respect to 𝑦 of the function 𝑓 ( 𝑥 , 𝑦 , 𝑧 ) = ( 𝑥 + 2 𝑦 + 3 𝑧 ) l n .

  • A 𝑥 𝑥 + 2 𝑦 + 3 𝑧
  • B 3 𝑥 + 2 𝑦 + 3 𝑧
  • C 2 𝑥 + 2 𝑦 + 3 𝑧
  • D 2 𝑦 𝑥 + 2 𝑦 + 3 𝑧
  • E 1 𝑥 + 2 𝑦 + 3 𝑧

Q12:

Find the first partial derivative with respect to 𝑥 of the function 𝑓 ( 𝑥 , 𝑦 ) = 𝑦 𝑒 .

  • A 𝑦 𝑒
  • B 𝑦 𝑒 + 2 𝑦 𝑒
  • C 𝑦 𝑒
  • D 2 𝑦 𝑒
  • E 2 𝑦 𝑒

Q13:

Find the first partial derivative of the function 𝑓 ( 𝑥 , 𝑦 ) = 𝑒 𝑥 + 𝑦 with respect to 𝑥 .

  • A 𝑒 ( 𝑥 + 𝑦 )
  • B 𝑒 𝑥 + 𝑦 + 2 𝑦 ( 𝑥 + 𝑦 )
  • C 𝑒 ( 𝑥 + 𝑦 )
  • D 𝑒 𝑥 + 𝑦 2 𝑦 ( 𝑥 + 𝑦 )
  • E 1 ( 𝑥 + 𝑦 )

Q14:

Find the first partial derivative of the function 𝑓 ( 𝑥 , 𝑦 , 𝑧 ) = 𝑥 with respect to 𝑥 .

  • A 𝑥
  • B 𝑦 𝑧 𝑥
  • C l n 𝑦 𝑧 𝑥
  • D 𝑦 𝑧 𝑥

Q15:

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

  • A 𝑦 + 1 ( 𝑥 + 𝑦 )
  • B 𝑥 1 ( 𝑥 + 𝑦 )
  • C 𝑦 2 𝑥 𝑦 1 ( 𝑥 + 𝑦 )
  • D 𝑦 1 ( 𝑥 + 𝑦 )
  • E 𝑦 + 2 𝑥 𝑦 1 ( 𝑥 + 𝑦 )

Q16:

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

  • A 1 𝑦 + 1
  • B 𝑥 + 1 ( 𝑦 + 1 )
  • C 𝑥 + 1 ( 𝑦 + 1 )
  • D ( 𝑦 + 1 ) ( 𝑦 + 1 ) d d
  • E ( 𝑦 + 1 ) + ( 𝑦 + 1 ) d d

Q17:

Find the first partial derivative with respect to 𝑦 of 𝑓 ( 𝑥 , 𝑦 ) = 𝑥 ( 𝑥 + 𝑦 ) .

  • A 2 𝑥 ( 𝑥 + 𝑦 )
  • B 𝑦 + 3 𝑥 ( 𝑥 + 𝑦 )
  • C 2 𝑥 ( 𝑥 + 𝑦 )
  • D 𝑦 𝑥 ( 𝑥 + 𝑦 )
  • E 2 𝑥 ( 𝑥 + 𝑦 )

Q18:

Find the first partial derivative with respect to 𝑥 of the function 𝑓 ( 𝑥 , 𝑦 ) = 𝑥 + 𝑦 + 4 .

  • A 𝑥 + 𝑥 + 𝑦 + 4 d d
  • B 2 𝑥 𝑥 + 𝑦 + 4
  • C 2 𝑥 + 𝑥 + 𝑦 + 4 d d
  • D 𝑥 𝑥 + 𝑦 + 4
  • E 1 2 𝑥 + 𝑦 + 4

Q19:

Find the first partial derivative with respect to 𝑦 of 𝑓 ( 𝑥 , 𝑦 , 𝑧 ) = 𝑥 + 𝑦 𝑧 c o s .

  • A 𝑦 𝑧 𝑥 + 𝑦 𝑧 s i n c o s
  • B 2 𝑦 𝑧 𝑥 + 𝑦 𝑧 c o s c o s
  • C 𝑦 𝑧 2 𝑥 + 𝑦 𝑧 c o s c o s
  • D 𝑦 𝑧 𝑥 + 𝑦 𝑧 s i n c o s
  • E 𝑦 𝑧 𝑥 + 𝑦 𝑧 c o s c o s

Q20:

Find, with respect to 𝑥 , the first partial derivative of 𝑓 ( 𝑥 , 𝑦 , 𝑧 , 𝑡 ) = 𝛼 𝑥 + 𝛽 𝑦 𝛾 𝑧 + 𝛿 𝑡 .

  • A 𝛼 𝛾 𝑧 + 𝛿 𝑡
  • B 𝛼 𝛾 𝑧 + 𝛿 𝑡
  • C 2 𝛿 𝑡 𝛼 𝑥 + 𝛽 𝑦 𝛼 𝛾 𝑧 + 𝛿 𝑡 ( 𝛾 𝑧 + 𝛿 𝑡 )
  • D 𝛼 𝛾 𝑧 + 𝛿 𝑡 2 𝛿 𝑡 𝛼 𝑥 + 𝛽 𝑦 ( 𝛾 𝑧 + 𝛿 𝑡 )

Q21:

Find the first partial derivative of the function 𝑓 ( 𝑥 , 𝑦 , 𝑧 ) = 𝑦 ( 𝑥 + 2 𝑧 ) t a n with respect to 𝑥 .

  • A 𝑦 ( 𝑥 + 2 𝑧 ) c s c
  • B 𝑦 ( 𝑥 + 2 𝑧 ) s e c
  • C 𝑦 ( 𝑥 + 2 𝑧 ) s e c
  • D 𝑦 ( 𝑥 + 2 𝑧 ) c s c
  • E ( 𝑥 + 2 𝑧 ) s e c

Q22:

Find the first partial derivative of the function 𝑓 ( 𝑥 , 𝑦 , 𝑧 ) = 𝑥 𝑦 𝑒 with respect to 𝑥 .

  • A 2 𝑥 𝑦 𝑒
  • B 𝑦 𝑒 𝑥 𝑦 𝑧 𝑒
  • C 𝑦 𝑧 𝑒
  • D 𝑦 𝑧 𝑒
  • E 𝑦 𝑒 + 𝑥 𝑦 𝑧 𝑒

Q23:

Find the first partial derivative with respect to 𝑦 of 𝑓 ( 𝑥 , 𝑦 , 𝑧 ) = 𝑥 𝑦 𝑒 .

  • A 𝑥 𝑦 𝑒
  • B 2 𝑥 𝑦 𝑒
  • C 2 𝑥 𝑦 𝑒
  • D 𝑥 𝑦 𝑒

Q24:

Find the first partial derivative with respect to 𝑦 of 𝑓 ( 𝑥 , 𝑦 , 𝑧 , 𝑡 ) = 𝑥 𝑦 𝑧 𝑡 c o s .

  • A 2 𝑥 𝑧 𝑡 c o s
  • B 2 𝑥 𝑧 𝑡 s i n
  • C 𝑥 𝑧 𝑡 s i n
  • D 𝑥 𝑧 𝑡 c o s

Q25:

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

  • A 𝑥 𝑦 𝑡 s i n
  • B 𝑥 𝑡 s i n
  • C 2 𝑥 𝑦 𝑡 s i n
  • D 𝑥 𝑡 s i n
  • E 𝑥 𝑦 𝑡 s i n

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