The partial derivative ∂w/∂y at (2,1,1) is -3/10.
Calculating ∂w/∂y using implicit differentiation
Here's how to calculate the partial derivative ∂w/∂y of the given equation at (x,y,w) = (2,1,1):
Rewrite the equation as a function of w:
Start by isolating w on one side. Multiply both sides of the equation by 10(w^2 + x^2) to get:
10(w^2 + x^2) + 10(u^2 + y^2) = 7w^2
Differentiate both sides implicitly with respect to y:
Treat w and u as constants while differentiating. This means their derivatives with respect to y will be zero. Take the derivative of both sides using the chain rule:
20w(∂w/∂y) + 20y = 14yw
Solve for ∂w/∂y:
Isolate ∂w/∂y:
∂w/∂y = (14yw - 20y) / 20w
Plug in the values at (x,y,w) = (2,1,1):
Substitute x = 2, y = 1, and w = 1:
∂w/∂y = (14 * 1 * 1 - 20 * 1) / (20 * 1) = -6 / 20 = -3/10
Therefore, the partial derivative ∂w/∂y at (2,1,1) is -3/10.