Can someone help me find partial derivatives?

Question

a) i) If x=cosu - e^-v, u=t^2 - s^2 and v=ts, find partial derivatives

∂x¦ and ∂x¦

∂t¦s ∂s¦t

ii) Given that x^2 + y^2 + z^2 = 1 find partial derivatives

∂z/∂x¦y and ∂z/∂y¦x

sorry bout how its written hope you can understand it, thank you

Answer 1

[a-i]

Using "d" for partial derivatives:

dx/dt = -sin(u)du/dt + e^(-v)dv/dt

du/dt = 2t ..... since s is constant

dv/dt = s ...... since s is constant

dx/dt = -2*t*sin(u) + s*e^(-v)

dx/ds = -sin(u)du/ds + e^(-v)dy/ds

du/ds = -2s ..... since in this case t is constant

dv/ds = t ......... again since t is constant

dx/ds = -2*s*sin(u) + t*e^(-v)

[a - ii]

2x + 2z(dz/dx) = 0

dz/dx = -x/z

2y + 2z(dz/dy) = 0

dz/dy = -y/z

z = SQRT(1 - x^2 - y^2)

Answer 2

Think of it this way, the variables that you aren't asked to find the derivative for are treated as constants.