Final answer:
To find the partial derivatives ∂z/∂s and ∂z/∂t using the chain rule, differentiate z = tan(uv) with respect to both u and v, then differentiate u and v with respect to s and t, and plug the derived expressions into the chain rule formula.
Step-by-step explanation:
The student's question involves using the chain rule in calculus to find partial derivatives of the function z with respect to s and t, given z as a function of u and v, where u and v themselves are functions of s and t.
First, we find ∂z/∂s by differentiating z with respect to s, using the chain rule:
dz/ds = (∂z/∂u)(du/ds) + (∂z/∂v)(dv/ds)
And for ∂z/∂t:
dz/dt = (∂z/∂u)(du/dt) + (∂z/∂v)(dv/dt)
To solve for these derivatives, we would differentiate tan(uv) with respect to u and v, and then differentiate u = 9s + 7t and v = 7s - 9t with respect to s and t. Subsequently, we would plug in the resulting expressions into the formulas above.