Final answer:
To find ∂²z/(∂r ∂s), we apply the chain rule multiple times considering the functions x = r² s² and y = 3rs. The process involves computing partial derivatives of x and y with respect to r and s, and then combining these with the partial derivatives of z with respect to x and y.
Step-by-step explanation:
To find the mixed second-order partial derivative ∂²z/(∂r ∂s) when z = f(x, y), where x = r² s² and y = 3rs, we need to use the chain rule for partial derivatives multiple times. First, we compute the first-order derivatives ∂z/∂x and ∂z/∂y. Then, we differentiate each of these with respect to r, while taking into account how x and y depend on r. Finally, we differentiate the resulting expressions with respect to s. The process involves calculating:
- ∂x/∂r = 2rs²
- ∂x/∂s = 2r²s
- ∂y/∂r = 3s
- ∂y/∂s = 3r
We apply the chain rule for each of these derivatives. The second-order mixed derivative is then found by summing the products of these derivatives and the corresponding second-order partial derivatives of z with respect to x and y.
The calculation is as follows:
- Compute ∂z/∂r = (∂z/∂x)(∂x/∂r) + (∂z/∂y)(∂y/∂r)
- Compute ∂z/∂s = (∂z/∂x)(∂x/∂s) + (∂z/∂y)(∂y/∂s)
- Finally, compute ∂(∂z/∂r)/∂s
It is important to note that this is a general procedure and without the explicit function f(x, y), we cannot compute the numerical value of the derivative.