Question

If x = K(s, m(s), p(s, t)), find expressions for ∂x/∂s and ∂x/∂t.

I got a different solution to another friend of mine.

Can somebody see if I am correct or not?

My solution:
∂x/∂s = (∂x/∂K)(dK/∂s)+(∂x/∂K)(∂K/∂m)(∂m/∂s)+(∂x/∂K)(∂k/∂p)(∂p/∂s)

∂x/∂t = (∂x/∂K)(∂K/∂p)(∂t/∂p)

Answer 1

$$\frac{\partial x}{\partial s}=\frac{\partial K}{\partial s}+\frac{\partial K}{\partial m}\cdot\frac{dm}{ds}+\frac{\partial K}{\partial p}\cdot\frac{\partial p}{\partial s}$$this is identical to yours since:$$\frac{dm}{ds}=\frac{\partial m}{\partial s}$$and$$\frac{\partial x}{\partial K}=1$$

Answer 2

similarly we have$$\frac{\partial x}{\partial t}=\frac{\partial K}{\partial p}\cdot\frac{\partial p}{\partial t}$$

Answer 3

Thanks so much for this - really appreciate it!
Can I ask how you know that ∂x/∂K = 1?

Answer 4

you said \(x=K\)

Answer 5

Nice. Hadn't really taken a step back and thought about that relationship fully.
Appreciate it a lot man, thanks again!

Answer 6

np glad I could help