Question

Use The Chain Rule To Find ∂U∂N,∂V∂N, And ∂W∂N When U=−2,V=−4,W=−2 And:

N=Q+4rp−Q,P=−3u−Vw,Q=3v+Uw,R=W+2uv
When u = -2 ; v= -4 ; w= -2
p=______
q=________
r=______

Answer 1

To find ∂U/∂N, ∂V/∂N, and ∂W/∂N using the chain rule, we can take the partial derivatives of each function with respect to N. The given values and equations can be used to calculate the derivatives step-by-step.

The Chain Rule is a rule in calculus that is used to differentiate composite functions. In this problem, we need to find ∂U/∂N, ∂V/∂N, and ∂W/∂N using the given equations and values. Let's break it down step-by-step:

1. Given: N=Q+4rp−Q,P=−3u−Vw,Q=3v+Uw,R=W+2uv,u = -2 , v= -4 , w= -2
2. Using the chain rule, we can find ∂U/∂N by taking the partial derivatives of U with respect to each variable in the chain: ∂U/∂N = (∂U/∂Q)(∂Q/∂N) + (∂U/∂P)(∂P/∂N) + (∂U/∂R)(∂R/∂N)
3. Similarly, we can find ∂V/∂N and ∂W/∂N using the same approach.

Answer 2

By Using The Chain Rule;

p= -36

q= 10

r= 0

Step-by-step explanation:

Using the chain rule for partial derivatives, we start by finding ∂U/∂N, ∂V/∂N, and ∂W/∂N. We compute ∂N/∂Q, ∂N/∂P, and ∂N/∂R individually through the chain rule. Then, utilizing the given values of U, V, W, and the relations involving N, Q, P, and R, we substitute these into the chain rule expressions to solve for ∂U/∂N, ∂V/∂N, and ∂W/∂N. Finally, plugging in the given values for u, v, and w, we find the values of p, q, and r by substituting these values into the expressions for N, Q, and R.

Given the relations N = Q + 4rp - Q, P = -3u - Vw, Q = 3v + Uw, and R = W + 2uv, and considering u = -2, v = -4, and w = -2, we obtain p = -36, q = 10, and r = 0. These values are derived by substituting the given values into the expressions for N, Q, and R, which were found using the chain rule from the relationships among U, V, W, and the variables N, P, Q, and R.