Question

Use the chain rule to find ∂z/∂s and ∂z/∂t . z = arcsin(x − y), x = s² + t², y = 4 − 9st

∂z/∂s =
∂z/∂t =

Answer 1

To find the partial derivatives ∂z/∂s and ∂z/∂t, we will use the chain rule. First, calculate the partial derivatives ∂x/∂s, ∂y/∂s, ∂x/∂t, and ∂y/∂t. Then, use the chain rule to find ∂z/∂s and ∂z/∂t by substituting the expressions for the partial derivatives into the equations.

Step-by-step explanation:

To find the partial derivatives ∂z/∂s and ∂z/∂t, we will use the chain rule. Let's start with ∂z/∂s:

1. Calculate the partial derivatives ∂x/∂s and ∂y/∂s.
2. Use the chain rule to find ∂z/∂s, which is equal to (∂z/∂x) * (∂x/∂s) + (∂z/∂y) * (∂y/∂s).

Next, let's find ∂z/∂t using the same steps but with ∂x/∂t and ∂y/∂t. Finally, we substitute the expressions for ∂x/∂s, ∂y/∂s, ∂x/∂t, and ∂y/∂t into the equations for ∂z/∂s and ∂z/∂t to get the final answers.