let
w=5xy-5yz+4xz, x=st, y=e^st , z= t^2
Compute
aw/as(2,5)=
aw/at(2,5)=
w=5xy-5yz+4xz, x=st, y=e^st , z= t^2
Compute
aw/as(2,5)=
aw/at(2,5)=
-
By the Chain Rule,
∂w/∂s = ∂w/∂x ∂x/∂s + ∂w/∂y ∂y/∂s + ∂w/∂z ∂z/∂s
.........= (5y+4z) * t + (5x-5z) * te^(st) + (-5y+4x) * 0
∂w/∂t = ∂w/∂x ∂x/∂t + ∂w/∂y ∂y/∂t + ∂w/∂z ∂z/∂t
.........= (5y+4z) * s + (5x-5z) * se^(st) + (-5y+4x) * 2t.
--------------------
At (s,t) = (2,5), we have (x,y,z) = (10, e^10, 25) via substitution.
So, we obtain
∂w(2,5)/∂s = 500 - 350 e^10 and ∂w(2,5)/∂t = 600 - 190e^10.
I hope this helps!
∂w/∂s = ∂w/∂x ∂x/∂s + ∂w/∂y ∂y/∂s + ∂w/∂z ∂z/∂s
.........= (5y+4z) * t + (5x-5z) * te^(st) + (-5y+4x) * 0
∂w/∂t = ∂w/∂x ∂x/∂t + ∂w/∂y ∂y/∂t + ∂w/∂z ∂z/∂t
.........= (5y+4z) * s + (5x-5z) * se^(st) + (-5y+4x) * 2t.
--------------------
At (s,t) = (2,5), we have (x,y,z) = (10, e^10, 25) via substitution.
So, we obtain
∂w(2,5)/∂s = 500 - 350 e^10 and ∂w(2,5)/∂t = 600 - 190e^10.
I hope this helps!