Consider 𝑓(𝑥,𝑦)=𝑥^2+𝑥𝑦+𝑦^2 where 𝑥=8𝑠+2𝑡, 𝑦=4𝑠+8𝑡. Calculate all necessary derivatives needed by the chain rule, and then use the chain rule to compute ∂𝑓/∂𝑠 and ∂𝑓/∂𝑡

Question

Consider 𝑓(𝑥,𝑦)=𝑥²+𝑥𝑦+𝑦² where 𝑥=8𝑠+2𝑡, 𝑦=4𝑠+8𝑡. Calculate all necessary derivatives needed by the chain rule, and then use the chain rule to compute ∂𝑓/∂𝑠 and ∂𝑓/∂𝑡.

dz/dx=

dz/dy=

dx/ds=

dx/dt=

dy/ds=

dy/dt=

df/ds=

df/dt=-

Answer 1

𝑓(𝑥,𝑦)=𝑥²+𝑥𝑦+𝑦² where 𝑥=8𝑠+2𝑡, 𝑦=4𝑠+8𝑡

df/dx= 2x + y

df/dy= 2y + x

dx/ds= 8

dx/dt= 2

dy/ds= 4

dy/dt= 8

𝑓(𝑥,𝑦)=𝑥²+𝑥𝑦+𝑦^{2 Plug in x and y.}

ƒ(s,t) = (8s + 2t)^2 + (8s + 2t)(4s + 8t) +(4s + 8t)^2

= (64s²+32st+4t²)+(32s²+72st+16t²)+(16s²+64st+64t²)

= (112s²+168st+84t²)

df/ds= 224s + 168t

df/dt= 168t + 168s

using the chain rule:

df/ds= df/dx * dx/ds

= (2x + y)*8 [plug in x and y]

= (2(8s +2t) + (4s+8t))*8

=....

= 224s + 168t

The same process for df/dt which will get what was calculated above. Just thought I show that partial derivatives do match Both chain rule and the long hand method.