118. Chain Rule

𝑧=𝑓(𝑢,𝑣)𝑢=𝑓(𝑥,𝑦)𝑣=𝑓(𝑥,𝑦)

The derivative of 𝑧 with respect to 𝑥 is the sum of all possible paths from 𝑧 to 𝑥:

𝜕𝑧𝜕𝑥=𝜕𝑧𝜕𝑢𝜕𝑢𝜕𝑥+𝜕𝑧𝜕𝑣𝜕𝑣𝜕𝑥
Example

Let:

  • 𝑢=𝑥+𝑦

  • 𝑣=𝑥𝑦

  • 𝑧=𝑢2+sin(𝑣)

We want to compute:

𝜕𝑧𝜕𝑥

Step 1: Apply the multivariable chain rule

𝜕𝑧𝜕𝑥=𝜕𝑧𝜕𝑢𝜕𝑢𝜕𝑥+𝜕𝑧𝜕𝑣𝜕𝑣𝜕𝑥

Step 2: Compute partial derivatives

  • 𝜕𝑢𝜕𝑥=𝜕𝜕𝑥(𝑥+𝑦)=1+0=1

  • 𝜕𝑣𝜕𝑥=𝜕𝜕𝑥(𝑥𝑦)=1𝑦=𝑦

  • 𝜕𝑧𝜕𝑢=𝜕𝜕𝑢(𝑢2+sin(𝑣))=2𝑢+0=2𝑢

  • 𝜕𝑧𝜕𝑣=𝜕𝜕𝑣(𝑢2+sin(𝑣))=0+cos(𝑣)=cos(𝑣)

Step 3: Plug in:

𝜕𝑧𝜕𝑥=2𝑢⋅1+cos(𝑣)⋅𝑦=2(𝑥+𝑦)+𝑦cos(𝑥𝑦)