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Worksheet: Partial Derivatives

Q1:

Find the first partial derivative of the function 𝑓 ( π‘₯ ) = √ π‘₯ + 𝑦 βˆ’ 4 3 2 with respect to π‘₯ .

  • A 2 π‘₯  ( π‘₯ + 𝑦 βˆ’ 4 ) 3 2 2
  • B 1 3  ( π‘₯ + 𝑦 βˆ’ 4 ) 3 2 2
  • C 2 π‘₯ + 3  ( π‘₯ + 𝑦 βˆ’ 4 ) d d 𝑦 π‘₯ 2 2 3
  • D 2 π‘₯ 3  ( π‘₯ + 𝑦 βˆ’ 4 ) 3 2 2
  • E 2 π‘₯ +  ( π‘₯ + 𝑦 βˆ’ 4 ) d d 𝑦 π‘₯ 2 2 3

Q2:

Find the first partial derivative with respect to 𝑦 of 𝑓 ( π‘₯ , 𝑦 , 𝑧 ) = π‘₯ 𝑦 𝑧 + 2 𝑦 𝑧 3 2 .

  • A 3 π‘₯ 𝑦 𝑧 + π‘₯ 𝑧 + 2 π‘₯ 𝑦 𝑧 + 2 𝑧 + 2 𝑦 2 2 3 2 3
  • B 3 π‘₯ 𝑦 𝑧 2 2
  • C 2 π‘₯ 𝑦 𝑧 + 2 𝑦 3
  • D π‘₯ 𝑧 + 2 𝑧 3 2

Q3:

Find the first partial derivative with respect to π‘₯ of the function 𝑓 ( π‘₯ , 𝑦 ) = π‘₯ + 2 𝑦 .

Q4:

Find the first partial derivative with respect to 𝑦 of the function 𝑓 ( π‘₯ , 𝑦 ) = π‘₯ + 2 𝑦 .

Q5:

Find the first partial derivative with respect to π‘₯ of the function 𝑓 ( π‘₯ , 𝑦 ) = π‘₯ + 𝑦 2 2 .

  • A 2 π‘₯ + 2 𝑦
  • B 2 𝑦
  • C π‘₯ + 2 𝑦 2
  • D 2 π‘₯
  • E π‘₯

Q6:

Find the first partial derivative with respect to 𝑦 of the function 𝑓 ( π‘₯ , 𝑦 ) = π‘₯ βˆ’ 𝑦 + 6 π‘₯ 𝑦 + 4 π‘₯ βˆ’ 8 𝑦 + 2 2 2 .

  • A 2 ο€½ π‘₯ βˆ’ 𝑦 π‘₯ ( 𝑦 βˆ’ 3 π‘₯ + 4 ) + 2 𝑦 + 2  d d
  • B 2 ( π‘₯ + 3 𝑦 + 2 )
  • C 2 ( 4 π‘₯ βˆ’ 𝑦 βˆ’ 2 )
  • D 2 ( 3 π‘₯ βˆ’ 𝑦 βˆ’ 4 )
  • E 2 ( π‘₯ + 2 𝑦 βˆ’ 2 )

Q7:

Find the first partial derivative with respect to π‘₯ of the function 𝑓 ( π‘₯ , 𝑦 ) = π‘₯ + 5 π‘₯ 𝑦 4 3 .

  • A 4 π‘₯ + 5 π‘₯ 𝑦 4 3
  • B 1 5 π‘₯ 𝑦 2
  • C π‘₯ + 1 5 π‘₯ 𝑦 4 2
  • D 4 π‘₯ + 5 𝑦 3 3
  • E 4 π‘₯ + 1 5 π‘₯ 𝑦 + 5 𝑦 3 2 3

Q8:

Find the first partial derivative with respect to 𝑦 of 𝑓 ( π‘₯ , 𝑦 ) = π‘₯ 𝑦 βˆ’ 3 𝑦 2 4 .

  • A π‘₯ + 2 π‘₯ 𝑦 βˆ’ 1 2 𝑦 2 3
  • B 2 π‘₯ 𝑦
  • C π‘₯ 𝑦 βˆ’ 1 2 𝑦 2 4
  • D π‘₯ βˆ’ 1 2 𝑦 2 3
  • E 2 π‘₯ 𝑦 βˆ’ 3 𝑦 4

Q9:

Find the first partial derivative with respect to π‘₯ of the function 𝑓 ( π‘₯ , 𝑦 ) = 𝑒 + π‘₯ 𝑦   .

  • A ο€½ π‘₯ 𝑦 π‘₯ + 𝑦  ( 𝑒 + 1 ) d d  
  • B π‘₯ ( 𝑒 + 1 )  
  • C 𝑦 ( 𝑒 + 1 ) 
  • D 𝑦 ( 𝑒 + 1 )  
  • E π‘₯ ( 𝑒 + 1 ) 

Q10:

Find the first partial derivative with respect to 𝑦 of the function 𝑓 ( π‘₯ , 𝑦 ) = π‘₯ 4 .

  • A 4 π‘₯ 4
  • B 4 π‘₯ 3
  • C π‘₯ 3
  • D0
  • E 4 π‘₯ 𝑦 3

Q11:

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

  • A 3 π‘₯ + 2 𝑦 + 3 𝑧
  • B 1 π‘₯ + 2 𝑦 + 3 𝑧
  • C 2 𝑦 π‘₯ + 2 𝑦 + 3 𝑧
  • D 2 π‘₯ + 2 𝑦 + 3 𝑧
  • E π‘₯ π‘₯ + 2 𝑦 + 3 𝑧

Q12:

Find the first partial derivative with respect to π‘₯ of the function 𝑓 ( π‘₯ , 𝑦 ) = 𝑦 𝑒 2 βˆ’ π‘₯ .

  • A βˆ’ 𝑦 𝑒 + 2 𝑦 𝑒 2 βˆ’ π‘₯ βˆ’ π‘₯
  • B 2 𝑦 𝑒 βˆ’ π‘₯
  • C 𝑦 𝑒 2 βˆ’ π‘₯
  • D βˆ’ 𝑦 𝑒 2 βˆ’ π‘₯
  • E 2 𝑦 𝑒 2 βˆ’ π‘₯

Q13:

Find the first partial derivative of the function 𝑓 ( π‘₯ , 𝑦 ) = 𝑒 π‘₯ + 𝑦 𝑦 2 with respect to π‘₯ .

  • A 𝑒 ο€Ή π‘₯ + 𝑦 + 2 𝑦  ( π‘₯ + 𝑦 ) 𝑦 2 2 2
  • B 𝑒 ο€Ή π‘₯ + 𝑦 βˆ’ 2 𝑦  ( π‘₯ + 𝑦 ) 𝑦 2 2 2
  • C 𝑒 ( π‘₯ + 𝑦 ) 𝑦 2 2
  • D βˆ’ 𝑒 ( π‘₯ + 𝑦 ) 𝑦 2 2
  • E 1 ( π‘₯ + 𝑦 ) 2 2

Q14:

Find the first partial derivative with respect to π‘₯ of the function.

  • A π‘₯ 𝑦 𝑧 βˆ’ 1
  • B 𝑦 𝑧 π‘₯ 𝑦 𝑧
  • C l n ο€» 𝑦 𝑧  π‘₯ 𝑦 𝑧
  • D 𝑦 𝑧 π‘₯ 𝑦 𝑧 βˆ’ 1

Q15:

Find the first partial derivative of the function 𝑓 ( π‘₯ , 𝑦 ) = π‘₯ 𝑦 + 1 π‘₯ + 𝑦 with respect to π‘₯ .

  • A 𝑦 βˆ’ 2 π‘₯ 𝑦 βˆ’ 1 ( π‘₯ + 𝑦 ) 2 2
  • B π‘₯ βˆ’ 1 ( π‘₯ + 𝑦 ) 2 2
  • C 𝑦 + 2 π‘₯ 𝑦 βˆ’ 1 ( π‘₯ + 𝑦 ) 2 2
  • D 𝑦 βˆ’ 1 ( π‘₯ + 𝑦 ) 2 2
  • E 𝑦 + 1 ( π‘₯ + 𝑦 ) 2 2

Q16:

Find the first partial derivative of the function 𝑓 ( π‘₯ , 𝑦 ) = π‘₯ + 1 𝑦 + 1 with respect to π‘₯ .

  • A π‘₯ + 1 ( 𝑦 + 1 ) 
  • B βˆ’ π‘₯ + 1 ( 𝑦 + 1 ) 
  • C ( 𝑦 + 1 ) βˆ’ ( 𝑦 + 1 ) d d   
  • D 1 𝑦 + 1
  • E ( 𝑦 + 1 ) + ( 𝑦 + 1 ) d d   

Q17:

Find the first partial derivative with respect to 𝑦 of 𝑓 ( π‘₯ , 𝑦 ) = π‘₯ ( π‘₯ + 𝑦 ) 2 .

  • A 2 π‘₯ ( π‘₯ + 𝑦 ) 3
  • B 𝑦 βˆ’ π‘₯ ( π‘₯ + 𝑦 ) 3
  • C 𝑦 + 3 π‘₯ ( π‘₯ + 𝑦 ) 3
  • D βˆ’ 2 π‘₯ ( π‘₯ + 𝑦 ) 3
  • E βˆ’ 2 π‘₯ ( π‘₯ + 𝑦 ) 4

Q18:

Find the first partial derivative with respect to π‘₯ of the function 𝑓 ( π‘₯ , 𝑦 ) = √ π‘₯ + 𝑦 + 4  .

  • A 2 π‘₯ √ π‘₯ + 𝑦 + 4 
  • B 1 2 √ π‘₯ + 𝑦 + 4 
  • C 2 π‘₯ + √ π‘₯ + 𝑦 + 4 d d   
  • D π‘₯ √ π‘₯ + 𝑦 + 4 
  • E π‘₯ + √ π‘₯ + 𝑦 + 4      d d

Q19:

Find the first partial derivative with respect to 𝑦 of 𝑓 ( π‘₯ , 𝑦 , 𝑧 ) = √ π‘₯ + 𝑦 𝑧 4 2 c o s .

  • A 𝑦 𝑧 √ 2 π‘₯ + 𝑦 𝑧 c o s c o s 4 2
  • B 2 𝑦 𝑧 √ π‘₯ + 𝑦 𝑧 c o s c o s 4 2
  • C βˆ’ 𝑦 𝑧 √ π‘₯ + 𝑦 𝑧 s i n c o s 4 2
  • D 𝑦 𝑧 √ π‘₯ + 𝑦 𝑧 c o s c o s 4 2
  • E 𝑦 𝑧 √ π‘₯ + 𝑦 𝑧 s i n c o s 4 2

Q20:

Find, with respect to π‘₯ , the first partial derivative of 𝑓 ( π‘₯ , 𝑦 , 𝑧 , 𝑑 ) = 𝛼 π‘₯ + 𝛽 𝑦 𝛾 𝑧 + 𝛿 𝑑 2 2 .

  • A 𝛼 ο€Ή 𝛾 𝑧 + 𝛿 𝑑  βˆ’ 2 𝛿 𝑑 ο€Ή 𝛼 π‘₯ + 𝛽 𝑦  ( 𝛾 𝑧 + 𝛿 𝑑 ) 2 2 2 2
  • B βˆ’ 𝛼 𝛾 𝑧 + 𝛿 𝑑 2
  • C 2 𝛿 𝑑 ο€Ή 𝛼 π‘₯ + 𝛽 𝑦  βˆ’ 𝛼 ο€Ή 𝛾 𝑧 + 𝛿 𝑑  ( 𝛾 𝑧 + 𝛿 𝑑 ) 2 2 2 2
  • D 𝛼 𝛾 𝑧 + 𝛿 𝑑 2

Q21:

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

  • A 𝑦 ( π‘₯ + 2 𝑧 ) c s c 2
  • B βˆ’ 𝑦 ( π‘₯ + 2 𝑧 ) s e c 2
  • C βˆ’ 𝑦 ( π‘₯ + 2 𝑧 ) c s c 2
  • D 𝑦 ( π‘₯ + 2 𝑧 ) s e c 2
  • E βˆ’ ( π‘₯ + 2 𝑧 ) s e c 2

Q22:

Find the first partial derivative of the function 𝑓 ( π‘₯ , 𝑦 , 𝑧 ) = π‘₯ 𝑦 𝑒     with respect to π‘₯ .

  • A βˆ’ 𝑦 𝑧 𝑒    
  • B 𝑦 𝑒 + π‘₯ 𝑦 𝑧 𝑒        
  • C 𝑦 𝑧 𝑒    
  • D 𝑦 𝑒 βˆ’ π‘₯ 𝑦 𝑧 𝑒        
  • E 2 π‘₯ 𝑦 𝑒   

Q23:

Find the first partial derivative with respect to 𝑦 of 𝑓 ( π‘₯ , 𝑦 , 𝑧 ) = π‘₯ 𝑦 𝑒 2 βˆ’ π‘₯ 𝑧 .

  • A 2 π‘₯ 𝑦 𝑒 2 βˆ’ π‘₯ 𝑧
  • B π‘₯ 𝑦 𝑒 βˆ’ π‘₯ 𝑧
  • C βˆ’ π‘₯ 𝑦 𝑒 2 2 βˆ’ π‘₯ 𝑧
  • D 2 π‘₯ 𝑦 𝑒 βˆ’ π‘₯ 𝑧

Q24:

Find the first partial derivative with respect to 𝑦 of 𝑓 ( π‘₯ , 𝑦 , 𝑧 , 𝑑 ) = π‘₯ 𝑦 ο€» 𝑧 𝑑  2 c o s .

  • A βˆ’ π‘₯ ο€» 𝑧 𝑑  2 s i n
  • B 2 π‘₯ ο€» 𝑧 𝑑  c o s
  • C βˆ’ 2 π‘₯ ο€» 𝑧 𝑑  s i n
  • D π‘₯ ο€» 𝑧 𝑑  2 c o s

Q25:

Find the first partial derivative with respect to 𝑧 of the function 𝑓 ( π‘₯ , 𝑦 , 𝑧 , 𝑑 ) = π‘₯ 𝑦 ο€» 𝑧 𝑑  2 c o s .

  • A βˆ’ π‘₯ ο€»  𝑑 2 𝑧 𝑑 s i n
  • B π‘₯ 𝑦 ο€»  𝑑 2 𝑧 𝑑 s i n
  • C π‘₯ ο€»  𝑑 2 𝑧 𝑑 s i n
  • D βˆ’ π‘₯ 𝑦 ο€»  𝑑 2 𝑧 𝑑 s i n
  • E βˆ’ 2 π‘₯ 𝑦 ο€»  𝑑 s i n 𝑧 𝑑