Worksheet: Partial Derivatives

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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.

  • A2𝑥3(𝑥+𝑦−4)
  • B13(𝑥+𝑦−4)
  • C2𝑥+3(𝑥+𝑦−4)dd
  • D2𝑥+(𝑥+𝑦−4)dd
  • E2𝑥(𝑥+𝑦−4)

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)=𝑥−𝑦+6𝑥𝑦+4𝑥−8𝑦+2.

  • A2(3𝑥−𝑦−4)
  • B2(𝑥+2𝑦−2)
  • C2𝑥−𝑦𝑥(𝑦−3𝑥+4)+2𝑦+2dd
  • D2(4𝑥−𝑦−2)
  • 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)=𝑥𝑦−3𝑦.

  • A𝑥−12𝑦
  • B2𝑥𝑦
  • C𝑥𝑦−12𝑦
  • D2𝑥𝑦−3𝑦
  • E𝑥+2𝑥𝑦−12𝑦

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.

Q11:

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

  • A𝑥𝑥+2𝑦+3𝑧
  • B1𝑥+2𝑦+3𝑧
  • C2𝑦𝑥+2𝑦+3𝑧
  • D2𝑥+2𝑦+3𝑧
  • E3𝑥+2𝑦+3𝑧

Q12:

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

  • A−𝑦𝑒+2𝑦𝑒
  • B𝑦𝑒
  • C2𝑦𝑒
  • D−𝑦𝑒
  • E2𝑦𝑒

Q13:

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

  • A−𝑒(𝑥+𝑦)
  • B𝑒(𝑥+𝑦)
  • C𝑒𝑥+𝑦−2𝑦(𝑥+𝑦)
  • D1(𝑥+𝑦)
  • E𝑒𝑥+𝑦+2𝑦(𝑥+𝑦)

Q14:

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

  • A𝑦𝑧𝑥
  • Bln𝑦𝑧𝑥
  • C𝑦𝑧𝑥
  • D𝑥

Q15:

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

  • A𝑦−1(𝑥+𝑦)
  • B𝑦+1(𝑥+𝑦)
  • C𝑦−2𝑥𝑦−1(𝑥+𝑦)
  • D𝑦+2𝑥𝑦−1(𝑥+𝑦)
  • E𝑥−1(𝑥+𝑦)

Q16:

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

  • A𝑥+1(𝑦+1)
  • B(𝑦+1)+(𝑦+1)dd
  • C−𝑥+1(𝑦+1)
  • D1𝑦+1
  • E(𝑦+1)−(𝑦+1)dd

Q17:

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

  • A𝑦−𝑥(𝑥+𝑦)
  • B𝑦+3𝑥(𝑥+𝑦)
  • C2𝑥(𝑥+𝑦)
  • D−2𝑥(𝑥+𝑦)
  • E−2𝑥(𝑥+𝑦)

Q18:

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

  • A𝑥√𝑥+𝑦+4
  • B2𝑥√𝑥+𝑦+4
  • C12√𝑥+𝑦+4
  • D2𝑥+√𝑥+𝑦+4dd
  • E𝑥+√𝑥+𝑦+4dd

Q19:

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

  • A𝑦𝑧√2𝑥+𝑦𝑧coscos
  • B𝑦𝑧√𝑥+𝑦𝑧coscos
  • C−𝑦𝑧√𝑥+𝑦𝑧sincos
  • D2𝑦𝑧√𝑥+𝑦𝑧coscos
  • E𝑦𝑧√𝑥+𝑦𝑧sincos

Q20:

Find, with respect to 𝑥, the first partial derivative of 𝑓(𝑥,𝑦,𝑧,𝑡)=𝛼𝑥+𝛽𝑦𝛾𝑧+𝛿𝑡.

  • A2𝛿𝑡𝛼𝑥+𝛽𝑦−𝛼𝛾𝑧+𝛿𝑡(𝛾𝑧+𝛿𝑡)
  • B𝛼𝛾𝑧+𝛿𝑡
  • C𝛼𝛾𝑧+𝛿𝑡−2𝛿𝑡𝛼𝑥+𝛽𝑦(𝛾𝑧+𝛿𝑡)
  • D−𝛼𝛾𝑧+𝛿𝑡

Q21:

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

  • A𝑦(𝑥+2𝑧)csc
  • B−𝑦(𝑥+2𝑧)sec
  • C−(𝑥+2𝑧)sec
  • D−𝑦(𝑥+2𝑧)csc
  • E𝑦(𝑥+2𝑧)sec

Q22:

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

  • A−𝑦𝑧𝑒
  • B𝑦𝑒−𝑥𝑦𝑧𝑒
  • C2𝑥𝑦𝑒
  • D𝑦𝑧𝑒
  • E𝑦𝑒+𝑥𝑦𝑧𝑒

Q23:

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

  • A2𝑥𝑦𝑒
  • B𝑥𝑦𝑒
  • C2𝑥𝑦𝑒
  • D−𝑥𝑦𝑒

Q24:

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

  • A𝑥𝑧𝑡cos
  • B−𝑥𝑧𝑡sin
  • C−2𝑥𝑧𝑡sin
  • D2𝑥𝑧𝑡cos

Q25:

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

  • A−𝑥𝑡sin
  • B𝑥𝑦𝑡sin
  • C−𝑥𝑦𝑡sin
  • D𝑥𝑡sin
  • E−2𝑥𝑦𝑡sin

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