Final answer:
To find ∂w/∂v when u=0 and v=0, use the chain rule to partially differentiate w with respect to v after substituting the expressions for x and y. Follow the differentiation with the evaluation of the resulting expression at the specified values of u and v.
Step-by-step explanation:
To find ∂w/∂v when u=0 and v=0, given w=x²+y/x, x=6u-v+1, and y=4u+4v-3, we need to compute the partial derivatives of w with respect to v through the chain rule.
First, let's substitute x and y into the equation for w:
w = (6u-v+1)² + (4u+4v-3)/(6u-v+1).
Now, we calculate ∂w/∂v by differentiating w with respect to v, while treating u as a constant.
Steps:
Differentiate (6u-v+1)² with respect to v: -2(6u-v+1).
Differentiate (4u+4v-3) with respect to v: 4.
Use the quotient rule to differentiate (4u+4v-3)/(6u-v+1) with respect to v.
Combine the results and substitute u=0, v=0.
The result is the value of ∂w/∂v evaluated at the given point.