Final answer:
To find the indicated partial derivatives of R = ln(u^2 + v^2 + w^2), we apply the Chain Rule by first differentiating R with respect to u, v, and w, then by differentiating u, v, and w with respect to x and y and substituting the values x = y = 2.
Step-by-step explanation:
To solve the given question, we need to use the Chain Rule for partial differentiation. The function R given by R = ln(u^2 + v^2 + w^2) is a composition of the functions u(x, y), v(x, y), and w(x, y). To find the partial derivatives ∂R/∂x and ∂R/∂y, we will differentiate R with respect to u, v, and w, and then use the chain rule to differentiate u, v, and w with respect to x and y respectively.
The partial derivative of R with respect to x is given by:
∂R/∂x = (∂R/∂u)(∂u/∂x) + (∂R/∂v)(∂v/∂x) + (∂R/∂w)(∂w/∂x)
Similarly, the partial derivative of R with respect to y is given by:
∂R/∂y = (∂R/∂u)(∂u/∂y) + (∂R/∂v)(∂v/∂y) + (∂R/∂w)(∂w/∂y)
Next, we will calculate each of the partial derivatives needed for the chain rule and substitute the given values of x and y as 2 to find the required results.