Question

Use the Chain Rule to find the indicated partial derivatives. R = ln(u^2 + v^2 + w^2), u = x + 9y, v = 5x - y, w = 5xy; partial differential R/partial differential x, partial differential R/partial differential y when x = y = 2 partial differential R/partial differential x = partial differential R/partial differential y =

Answer 1

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.