Worksheet: Implicit Differentiation and Partial derivatives

In this worksheet, we will practice finding the partial derivative of a function defined implicitly in two or more variables by a given equation.

Q1:

Find 𝑓, where 𝑓(𝑥,𝑦)=𝑥𝑦𝑥𝑦ln.

  • A𝑥𝑥𝑦1(𝑥𝑦)lnln
  • B𝑦𝑥𝑦𝑦(𝑥𝑦)lnln
  • C𝑥𝑥𝑦+𝑥(𝑥𝑦)lnln
  • D𝑥𝑥𝑦𝑥(𝑥𝑦)lnln
  • E𝑥𝑥𝑦𝑥𝑥𝑦lnln

Q2:

Find dd𝑥𝑦 when 𝑓(𝑥,𝑦)=𝑥+𝑥𝑦=0.

  • A3𝑥𝑦1+2𝑥𝑦
  • B3𝑥𝑦1+2𝑥𝑦
  • C1+2𝑥𝑦3𝑥𝑦
  • D1+2𝑥𝑦3𝑥𝑦

Q3:

Given that sin(𝑥𝑦𝑧)=𝑥𝑦𝑧, find 𝜕𝑧𝜕𝑦.

  • A1(𝑥𝑦𝑧)1+(𝑥𝑦𝑧)coscos
  • B𝑧+𝑧(𝑥𝑦𝑧)𝑦𝑦(𝑥𝑦𝑧)coscos
  • C𝑧𝑧(𝑥𝑦𝑧)𝑦+𝑦(𝑥𝑦𝑧)coscos
  • D𝑧𝑦𝑧(𝑥𝑦𝑧)𝑦+𝑥𝑦(𝑥𝑦𝑧)coscos

Q4:

Find 𝑓, where 𝑓(𝑥,𝑦)=𝑒.

  • A2𝑦𝑒
  • B(2𝑥+2𝑦)𝑒
  • C2𝑥𝑒
  • D2𝑒
  • E2𝑦𝑒

Q5:

Find 𝑓, where 𝑓(𝑥,𝑦)=𝑥𝑦𝑥+𝑦.

  • A𝑦𝑥𝑦(𝑥+𝑦)
  • B𝑦𝑥𝑦(𝑥+𝑦)
  • C𝑦𝑥𝑦𝑥+𝑦
  • D𝑦+3𝑥𝑦(𝑥+𝑦)

Q6:

Given that sin(𝑥𝑦𝑧)=𝑥+5𝑧+𝑦, find 𝜕𝑧𝜕𝑥.

  • A1𝑦𝑧(𝑥𝑦𝑧)5+𝑥𝑦(𝑥𝑦𝑧)coscos
  • B1𝑦𝑧(𝑥𝑦𝑧)5+𝑥𝑦(𝑥𝑦𝑧)coscos
  • C1𝑥𝑦𝑧(𝑥𝑦𝑧)5+𝑥𝑦𝑧(𝑥𝑦𝑧)coscos
  • D1𝑥𝑦𝑧(𝑥𝑦𝑧)5+𝑥𝑦𝑧(𝑥𝑦𝑧)coscos

Q7:

Find 𝑓, where 𝑓(𝑥,𝑦)=1𝑥𝑦.

  • A𝑦21𝑥𝑦
  • B11𝑥𝑦
  • C𝑥+𝑦1𝑥𝑦
  • D𝑦1𝑥𝑦
  • E𝑦1𝑥𝑦

Q8:

Given that 𝑥+𝑦+𝑧=1cos, find 𝜕𝑧𝜕𝑦.

  • A𝑦2𝑥3𝑧sin
  • Bsin𝑦3𝑧
  • C𝑦3𝑧sin
  • Dsin𝑦2𝑥3𝑧

Q9:

Given that 𝑥+𝑦+𝑧=1cos, find 𝜕𝑧𝜕𝑥.

  • A2𝑥3𝑧
  • B12𝑥3𝑧
  • C12𝑥3𝑧
  • D2𝑥3𝑧

Q10:

Find dd𝑦𝑥 when 𝑓(𝑥,𝑦)=𝑥𝑥𝑦=0.

  • A12𝑥𝑦3𝑥𝑦
  • B2𝑥𝑦3𝑥𝑦
  • C12𝑥𝑦3𝑥𝑦
  • D3𝑥𝑦1+2𝑥𝑦

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