Question

Use the Chain Rule to find the indicated partial derivatives.

u = r^2 + s^2
r = y + x cos t, s = x + y sin t

Answer 1

To find the indicated partial derivatives using the Chain Rule, we need to differentiate u with respect to r and s respectively. The partial derivative of u with respect to r is 2r + 2s(dr/dt) and with respect to s is r^2 + r(ds/dt).

Step-by-step explanation:

The given problem involves using the Chain Rule to find partial derivatives. The expression for u, r, and s are given. To find the indicated partial derivatives, we need to apply the Chain Rule. Here is the step-by-step explanation:

1. First, we find the partial derivative of u with respect to r. Using the Chain Rule, we differentiate u with respect to r, and then differentiate r with respect to t. This gives us the partial derivative of u with respect to r as 2r + 2s(dr/dt).
2. Next, we find the partial derivative of u with respect to s. Using the Chain Rule, we differentiate u with respect to s, and then differentiate s with respect to t. This gives us the partial derivative of u with respect to s as r^2 + r(ds/dt).

So, the partial derivatives of u with respect to r and s are 2r + 2s(dr/dt) and r^2 + r(ds/dt) respectively.