# Thread: Hard problem on the chain rule for multivariable function ??

1. ## Hard problem on the chain rule for multivariable function ??

Hi ..
this is a problem from old exam ..
am lost .. i cant do anything!
i know the chain rule for F(x,y) where x=g(s,t) and y = h(s,t)
or for f(x,y,z,...) and x=g(s,t,...) y = h(s,t,....) and z=m(s,t,....) .. etc
what am gonna to said as am good !
but am lost at this

Problem :
Let F(s,t) be a differentiable function, let G(x,y) = F(x+y,xy),
calculate G_x(2,3) and G_y(2,3) if F_s(2,3)=1, F_s(5,6)=1
and F_t(2,3) = 4 and F_t(5,6) = 7

G_x = differentiating G with respect to x
F_t = differentiating F with respect to t
and so on..

2. Originally Posted by TWiX
Hi ..
this is a problem from old exam ..
am lost .. i cant do anything!
i know the chain rule for F(x,y) where x=g(s,t) and y = h(s,t)
or for f(x,y,z,...) and x=g(s,t,...) y = h(s,t,....) and z=m(s,t,....) .. etc
what am gonna to said as am good !
but am lost at this

Problem :
Let F(s,t) be a differentiable function, let G(x,y) = F(x+y,xy),
calculate G_x(2,3) and G_y(2,3) if F_s(2,3)=1, F_s(5,6)=1
and F_t(2,3) = 4 and F_t(5,6) = 7

G_x = differentiating G with respect to x
F_t = differentiating F with respect to t
and so on..
Let $\displaystyle s = x+y$ and $\displaystyle t = xy$

By the chain rule

$\displaystyle G_x = F_s s_x + F_t t_x = F_s + y F_t$

then let $\displaystyle x = 2$ and $\displaystyle y = 3$. Similarly for $\displaystyle G_y$ .