Final answer:
To find ∂z/∂s and ∂z/∂t, we can use the chain rule. The derivative of z with respect to s is 8xsin(y) - 4tx²cos(y), and the derivative of z with respect to t is 8xsin(y) - 4sx²cos(y).
Step-by-step explanation:
To find ∂z/∂s and ∂z/∂t using the chain rule, we need to differentiate z with respect to x, y, s, and t. Let's start with ∂z/∂s:
∂z/∂s = (∂z/∂x)(∂x/∂s) + (∂z/∂y)(∂y/∂s)
Substituting the values of x, y, and z into the equation and differentiating:
∂z/∂s = (2xsin(y))(4s) + (x²cos(y))(-4t)
Simplifying the equation, we get ∂z/∂s = 8xsin(y) - 4tx²cos(y).
Now, let's find ∂z/∂t using the same process:
∂z/∂t = (∂z/∂x)(∂x/∂t) + (∂z/∂y)(∂y/∂t)
Substituting the values of x, y, and z into the equation and differentiating:
∂z/∂t = (2xsin(y))(4t) + (x²cos(y))(-4s)
Simplifying the equation, we get ∂z/∂t = 8xsin(y) - 4sx²cos(y).