Consider f(x,y) = tan^-1(xy) where x=u^2+v^2 and y=u^2+4v^2.
Use the chain rule to compute ∂z/∂u and ∂z/∂v in terms of u and v.
Assume ( )' is partial derivative operator shorthand.
x = u^2+v^2
y = u^2+4v^2
z = f(x,y) = tan^-1(xy) = arctan(xy)
∂x / ∂u = ∂(u^2+v^2) / ∂u
= ∂(u^2) / ∂u + ∂(v^2) / ∂u
= 2u + 0 (0 since we treat v like a constant for partial derivatives)
= 2u
∂y / ∂u = ∂(u^2+4v^2) / ∂u
= ∂(u^2) / ∂u + ∂(4v^2) / ∂u
= 2u + 0 (0 since we treat 4v^2 like a constant for partial derivatives)
= 2u
∂x / ∂v = ∂(u^2+v^2) / ∂v
= ∂(u^2) / ∂v + ∂(v^2) / ∂v
= 0 + 2v (0 since we treat v like a constant for partial derivatives)
= 2v
∂y / ∂v = ∂(u^2+4v^2) / ∂v
= ∂(u^2) / ∂v + ∂(4v^2) / ∂v
= 0 + 4*2*v (0 since we treat 4v^2 like a constant for partial derivatives)
= 8v
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1) Now to solve for ∂z/∂u using the chain rule:
∂z/∂u = ∂(arctan(xy)) / ∂u
∂z/∂u = d(arctan(xy))/du * ∂(xy)/∂u
Expand using the product rule.
= ∂(arctan(xy))/∂u * (x ∂y/∂u + y ∂x/∂u)
= 1 / (1 + (xy)^2) * (x ∂y/∂u + y ∂x/∂u)
= 1 / (1 + (xy)^2) * (x 2u + y 2u)
= 2u / (1 + (xy)^2) * (x + y)
x + y = (u^2+v^2) + (u^2+4v^2)
= 2u^2 + 5v^2
xy = (u^2+v^2) * (u^2+4v^2)
xy = u^4 + 5u^2v^2 + 4v^4
= [2u / (1 + (xy)^2)] * (x + y)
= [2u / (1 + (u^4 + 5u^2v^2 + 4v^4)^2)] * [(2u^2 + 5v^2)]
= [[2u*(2u^2 + 5v^2)] / [1 + (u^4 + 5u^2v^2 + 4v^4)^2)]
∂z/∂u = [4u^3 + 10uv^2] / [1 + (u^4 + 5u^2v^2 + 4v^4)^2]
I believe this is simplified enough and is in terms of u and v only.
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2) Now to solve for ∂z/∂v using the chain rule:
∂z/∂v = ∂(arctan(xy))/∂v
∂z/∂v = d(arctan(xy))/dv * ∂(xy)/∂v
Expand using the product rule.
= d(arctan(xy))/dv * (x ∂y/∂v + y ∂x/∂v)
= 1 / (1 + (xy)^2) * (x ∂y/∂v + y ∂x/∂v)
= 1 / (1 + (xy)^2) * (x 8v + y 2v)
= 2v / (1 + (xy)^2) * (4x + y)
4x + y = 4(u^2 + v^2) + (u^2 + 4v^2)
= 5u^2 + 8v^2
xy = u^4 + 5u^2v^2 + 4v^4
same as above
= [2v / (1 + (xy)^2)] * (4x + y)
= [2v / (1 + (u^4 + 5u^2v^2 + 4v^4)^2)] * [5u^2 + 8v^2]
= [2v * (5u^2 + 8v^2)] / [1 + (u^4 + 5u^2v^2 + 4v^4)^2)]
= [10u^2v + 16v^3)] / [1 + (u^4 + 5u^2v^2 + 4v^4)^2)]
∂z/∂v = = [10u^2v + 16v^3)] / [1 + (u^4 + 5u^2v^2 + 4v^4)^2)]
