Lesson Worksheet: Partial Derivatives Mathematics

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

Q1:

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

  • A2π‘₯3(π‘₯+π‘¦βˆ’4)
  • B13(π‘₯+π‘¦βˆ’4)
  • C2π‘₯+3(π‘₯+π‘¦βˆ’4)ddο˜ο—οŠ¨οŠ¨οŽ’
  • D2π‘₯+(π‘₯+π‘¦βˆ’4)ddο˜ο—οŠ¨οŠ¨οŽ’
  • E2π‘₯(π‘₯+π‘¦βˆ’4)

Q2:

Find the first partial derivative with respect to 𝑦 of 𝑓(π‘₯,𝑦,𝑧)=π‘₯𝑦𝑧+2π‘¦π‘§οŠ©οŠ¨.

  • A2π‘₯𝑦𝑧+2π‘¦οŠ©
  • B3π‘₯π‘¦π‘§οŠ¨οŠ¨
  • Cπ‘₯𝑧+2π‘§οŠ©οŠ¨
  • D3π‘₯𝑦𝑧+π‘₯𝑧+2π‘₯𝑦𝑧+2𝑧+2π‘¦οŠ¨οŠ¨οŠ©οŠ¨οŠ©

Q3:

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

Q4:

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

Q5:

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

  • A2𝑦
  • B2π‘₯
  • C2π‘₯+2𝑦
  • Dπ‘₯
  • Eπ‘₯+2π‘¦οŠ¨

Q6:

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

  • A2(3π‘₯βˆ’π‘¦βˆ’4)
  • B2(π‘₯+2π‘¦βˆ’2)
  • C2ο€½π‘₯βˆ’π‘¦π‘₯(π‘¦βˆ’3π‘₯+4)+2𝑦+2dd
  • D2(4π‘₯βˆ’π‘¦βˆ’2)
  • E2(π‘₯+3𝑦+2)

Q7:

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

  • Aπ‘₯+15π‘₯𝑦οŠͺ
  • B4π‘₯+5π‘₯𝑦οŠͺ
  • C4π‘₯+5π‘¦οŠ©οŠ©
  • D15π‘₯π‘¦οŠ¨
  • E4π‘₯+15π‘₯𝑦+5π‘¦οŠ©οŠ¨οŠ©

Q8:

Find the first partial derivative with respect to 𝑦 of 𝑓(π‘₯,𝑦)=π‘₯π‘¦βˆ’3π‘¦οŠ¨οŠͺ.

  • Aπ‘₯βˆ’12π‘¦οŠ¨οŠ©
  • B2π‘₯𝑦
  • Cπ‘₯π‘¦βˆ’12π‘¦οŠ¨οŠͺ
  • D2π‘₯π‘¦βˆ’3𝑦οŠͺ
  • Eπ‘₯+2π‘₯π‘¦βˆ’12π‘¦οŠ¨οŠ©

Q9:

Find the first partial derivative with respect to π‘₯ of the function 𝑓(π‘₯,𝑦)=𝑒+π‘₯π‘¦ο—ο˜.

  • Aπ‘₯(𝑒+1)
  • Bπ‘₯(𝑒+1)ο—ο˜
  • Cο€½π‘₯𝑦π‘₯+𝑦(𝑒+1)ddο—ο˜
  • D𝑦(𝑒+1)ο—ο˜
  • E𝑦(𝑒+1)

Q10:

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

Q11:

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

  • Aπ‘₯π‘₯+2𝑦+3𝑧
  • B1π‘₯+2𝑦+3𝑧
  • C2𝑦π‘₯+2𝑦+3𝑧
  • D2π‘₯+2𝑦+3𝑧
  • E3π‘₯+2𝑦+3𝑧

Q12:

Find the first partial derivative with respect to π‘₯ of the function 𝑓(π‘₯,𝑦)=π‘¦π‘’οŠ¨οŠ±ο—.

  • Aβˆ’π‘¦π‘’+2π‘¦π‘’οŠ¨οŠ±ο—οŠ±ο—
  • Bπ‘¦π‘’οŠ¨οŠ±ο—
  • C2π‘¦π‘’οŠ±ο—
  • Dβˆ’π‘¦π‘’οŠ¨οŠ±ο—
  • E2π‘¦π‘’οŠ¨οŠ±ο—

Q13:

Find the first partial derivative of the function 𝑓(π‘₯,𝑦)=𝑒π‘₯+π‘¦ο˜οŠ¨ with respect to π‘₯.

  • Aβˆ’π‘’(π‘₯+𝑦)
  • B𝑒(π‘₯+𝑦)
  • C𝑒π‘₯+π‘¦βˆ’2𝑦(π‘₯+𝑦)
  • D1(π‘₯+𝑦)
  • E𝑒π‘₯+𝑦+2𝑦(π‘₯+𝑦)

Q14:

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

  • A𝑦𝑧π‘₯ο‘‘ο‘’
  • Bln𝑦𝑧π‘₯ο‘‘ο‘’
  • C𝑦𝑧π‘₯ο‘‘ο‘’οŠ±οŠ§
  • Dπ‘₯ο‘‘ο‘’οŠ±οŠ§

Q15:

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

  • Aπ‘¦βˆ’1(π‘₯+𝑦)
  • B𝑦+1(π‘₯+𝑦)
  • Cπ‘¦βˆ’2π‘₯π‘¦βˆ’1(π‘₯+𝑦)
  • D𝑦+2π‘₯π‘¦βˆ’1(π‘₯+𝑦)
  • Eπ‘₯βˆ’1(π‘₯+𝑦)

Q16:

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

  • Aπ‘₯+1(𝑦+1)
  • B(𝑦+1)+(𝑦+1)ddο˜ο—οŠ¨
  • Cβˆ’π‘₯+1(𝑦+1)
  • D1𝑦+1
  • E(𝑦+1)βˆ’(𝑦+1)ddο˜ο—οŠ¨

Q17:

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

  • Aπ‘¦βˆ’π‘₯(π‘₯+𝑦)
  • B𝑦+3π‘₯(π‘₯+𝑦)
  • C2π‘₯(π‘₯+𝑦)
  • Dβˆ’2π‘₯(π‘₯+𝑦)
  • Eβˆ’2π‘₯(π‘₯+𝑦)οŠͺ

Q18:

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

  • Aπ‘₯√π‘₯+𝑦+4
  • B2π‘₯√π‘₯+𝑦+4
  • C12√π‘₯+𝑦+4
  • D2π‘₯+√π‘₯+𝑦+4ddο˜ο—οŠ¨
  • Eπ‘₯+√π‘₯+𝑦+4οŠ§οŠ¨ο˜ο—οŠ¨dd

Q19:

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

  • Aπ‘¦π‘§βˆš2π‘₯+𝑦𝑧coscosοŠͺ
  • Bπ‘¦π‘§βˆšπ‘₯+𝑦𝑧coscosοŠͺ
  • Cβˆ’π‘¦π‘§βˆšπ‘₯+𝑦𝑧sincosοŠͺ
  • D2π‘¦π‘§βˆšπ‘₯+𝑦𝑧coscosοŠͺ
  • Eπ‘¦π‘§βˆšπ‘₯+𝑦𝑧sincosοŠͺ

Q20:

Find, with respect to π‘₯, the first partial derivative of 𝑓(π‘₯,𝑦,𝑧,𝑑)=𝛼π‘₯+𝛽𝑦𝛾𝑧+π›Ώπ‘‘οŠ¨οŠ¨.

  • A2𝛿𝑑𝛼π‘₯+π›½π‘¦ο…βˆ’π›Όο€Ήπ›Ύπ‘§+𝛿𝑑(𝛾𝑧+𝛿𝑑)
  • B𝛼𝛾𝑧+π›Ώπ‘‘οŠ¨
  • C𝛼𝛾𝑧+π›Ώπ‘‘ο…βˆ’2𝛿𝑑𝛼π‘₯+𝛽𝑦(𝛾𝑧+𝛿𝑑)
  • Dβˆ’π›Όπ›Ύπ‘§+π›Ώπ‘‘οŠ¨

Q21:

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

  • A𝑦(π‘₯+2𝑧)csc
  • Bβˆ’π‘¦(π‘₯+2𝑧)sec
  • Cβˆ’(π‘₯+2𝑧)sec
  • Dβˆ’π‘¦(π‘₯+2𝑧)csc
  • E𝑦(π‘₯+2𝑧)sec

Q22:

Find the first partial derivative of the function 𝑓(π‘₯,𝑦,𝑧)=π‘₯π‘¦π‘’οŠ¨οŠ±ο—ο™ with respect to π‘₯.

  • Aβˆ’π‘¦π‘§π‘’οŠ¨οŠ±ο—ο™
  • Bπ‘¦π‘’βˆ’π‘₯π‘¦π‘§π‘’οŠ¨οŠ±ο—ο™οŠ¨οŠ±ο—ο™
  • C2π‘₯π‘¦π‘’οŠ±ο—ο™
  • Dπ‘¦π‘§π‘’οŠ¨οŠ±ο—ο™
  • E𝑦𝑒+π‘₯π‘¦π‘§π‘’οŠ¨οŠ±ο—ο™οŠ¨οŠ±ο—ο™

Q23:

Find the first partial derivative with respect to 𝑦 of 𝑓(π‘₯,𝑦,𝑧)=π‘₯π‘¦π‘’οŠ¨οŠ±ο—ο™.

  • A2π‘₯π‘¦π‘’οŠ¨οŠ±ο—ο™
  • Bπ‘₯π‘¦π‘’οŠ±ο—ο™
  • C2π‘₯π‘¦π‘’οŠ±ο—ο™
  • Dβˆ’π‘₯π‘¦π‘’οŠ¨οŠ¨οŠ±ο—ο™

Q24:

Find the first partial derivative with respect to 𝑦 of 𝑓(π‘₯,𝑦,𝑧,𝑑)=π‘₯π‘¦ο€»π‘§π‘‘ο‡οŠ¨cos.

  • Aπ‘₯ο€»π‘§π‘‘ο‡οŠ¨cos
  • Bβˆ’π‘₯ο€»π‘§π‘‘ο‡οŠ¨sin
  • Cβˆ’2π‘₯𝑧𝑑sin
  • D2π‘₯𝑧𝑑cos

Q25:

Find the first partial derivative with respect to 𝑧 of the function 𝑓(π‘₯,𝑦,𝑧,𝑑)=π‘₯π‘¦ο€»π‘§π‘‘ο‡οŠ¨cos.

  • Aβˆ’π‘₯ο€»ο‡π‘‘οŠ¨ο™οsin
  • Bπ‘₯π‘¦ο€»ο‡π‘‘οŠ¨ο™οsin
  • Cβˆ’π‘₯π‘¦ο€»ο‡π‘‘οŠ¨ο™οsin
  • Dπ‘₯ο€»ο‡π‘‘οŠ¨ο™οsin
  • Eβˆ’2π‘₯𝑦𝑑sin

Nagwa uses cookies to ensure you get the best experience on our website. Learn more about our Privacy Policy.