Final answer:
To find ∂z/∂s and ∂z/∂t, we rewrite z in terms of s and t and apply the chain rule to find the partial derivatives. ∂z/∂s = 3(s cos(t))²(s sin(t))⁹ + 9(s cos(t))³(s sin(t))⁸(sin(t)), and ∂z/∂t = 9(s cos(t))³(s sin(t))⁸(-s sin(t)).
Step-by-step explanation:
To find the partial derivatives ∂z/∂s and ∂z/∂t using the chain rule, we start by expressing z in terms of s and t. Since x = s cos(t) and y = s sin(t), we can rewrite z = x³y⁹ as z = (s cos(t))³(s sin(t))⁹.
Next, we take the partial derivative with respect to s by treating t as a constant. This gives us ∂z/∂s = 3(s cos(t))²(s sin(t))⁹ + 9(s cos(t))³(s sin(t))⁸(sin(t)).
Finally, we take the partial derivative with respect to t by treating s as a constant. This gives us ∂z/∂t = 9(s cos(t))³(s sin(t))⁸(-s sin(t)).