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(𝑥)=𝑥+𝑦4 with respect to 𝑥.

  • A2𝑥3(𝑥+𝑦4)
  • B13(𝑥+𝑦4)
  • C2𝑥+3(𝑥+𝑦4)dd
  • D2𝑥+(𝑥+𝑦4)dd
  • E2𝑥(𝑥+𝑦4)

Q2:

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

  • A2𝑥𝑦𝑧+2𝑦
  • B3𝑥𝑦𝑧
  • C𝑥𝑧+2𝑧
  • D3𝑥𝑦𝑧+𝑥𝑧+2𝑥𝑦𝑧+2𝑧+2𝑦

Q3:

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

Q4:

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

Q5:

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

  • A2𝑦
  • B2𝑥
  • C2𝑥+2𝑦
  • D𝑥
  • E𝑥+2𝑦

Q6:

Find the first partial derivative with respect to 𝑦 of the function 𝑓(𝑥,𝑦)=𝑥𝑦+6𝑥𝑦+4𝑥8𝑦+2.

  • A2(3𝑥𝑦4)
  • B2(𝑥+2𝑦2)
  • C2𝑥𝑦𝑥(𝑦3𝑥+4)+2𝑦+2dd
  • D2(4𝑥𝑦2)
  • E2(𝑥+3𝑦+2)

Q7:

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

  • A𝑥+15𝑥𝑦
  • B4𝑥+5𝑥𝑦
  • C4𝑥+5𝑦
  • D15𝑥𝑦
  • E4𝑥+15𝑥𝑦+5𝑦

Q8:

Find the first partial derivative with respect to 𝑦 of 𝑓(𝑥,𝑦)=𝑥𝑦3𝑦.

  • A𝑥12𝑦
  • B2𝑥𝑦
  • C𝑥𝑦12𝑦
  • D2𝑥𝑦3𝑦
  • E𝑥+2𝑥𝑦12𝑦

Q9:

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

  • A𝑥(𝑒+1)
  • B𝑥(𝑒+1)
  • C𝑥𝑦𝑥+𝑦(𝑒+1)dd
  • D𝑦(𝑒+1)
  • E𝑦(𝑒+1)

Q10:

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

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