Question

Suppose z=x^2*sin(y), x=2s^2−4t^2, y=4st.
A. Use the chain rule to find ∂z/∂s and ∂z/∂t as functions of x, y, s and t.

Answer 1

∂z/∂s = dz/dx * dx/ds + dz/dy * dy/ds
dz/dx = 2x sin(y)
dx/ds = 4s
dz/dy = x^2 cos(y)
dy/ds = 4t

∂z/∂s = 2x sin y * 4s + x^2 cos y * 4t = 8sx sin(y) + 4tx^2 cos(y)

∂z/∂t = dz/dx * dx/dt + dz/dy * dy/dt
dz/dx = 2x sin(y)
dx/dt = -8t
dz/dy = x^2 cos(y)
dy/dt = 4s

∂z/∂t = 2x sin y * -8t + x^2 cos y * 4s = -16tx sin(y) + 4sx^2 cos(y)