Final answer:
To find ∂z/∂s and ∂z/∂t for the function z = x⁷y⁹, first differentiate x and y with respect to s and t using the given transformation equations. Then use the chain rule to find ∂z/∂s and ∂z/∂t.
Step-by-step explanation:
To find ∂z/∂s and ∂z/∂t for the function z = x⁷y⁹, we first need to find ∂x/∂s, ∂x/∂t, ∂y/∂s, and ∂y/∂t using the given transformation equations.
From x = s cos(t), we can differentiate x with respect to s and t to obtain ∂x/∂s = cos(t) and ∂x/∂t = -s sin(t).
Similarly, from y = s sin(t), we can differentiate y with respect to s and t to obtain ∂y/∂s = sin(t) and ∂y/∂t = s cos(t).
Now, we can use the chain rule to find ∂z/∂s and ∂z/∂t:
∂z/∂s = (∂z/∂x) * (∂x/∂s) + (∂z/∂y) * (∂y/∂s) = 7x⁶y⁹ * cos(t) + x⁷ * sin(t)
∂z/∂t = (∂z/∂x) * (∂x/∂t) + (∂z/∂y) * (∂y/∂t) = 7x⁶y⁹ * (-s sin(t)) + x⁷ * (s cos(t))