Final answer:
Implicit differentiation of the equation 2x^2 + 5xy = 3 provides the derivative y'(3) as -0.76 when using x=3 and y=-10.
Step-by-step explanation:
To find y'(3) using implicit differentiation, we must differentiate both sides of the given equation 2x2 + 5xy = 3 with respect to x. When we do this, we respectively get 4x + 5y + 5xy' on the left side. This is because when we differentiate xy with respect to x, we need to use the Product Rule, which in this case gives us x(dy/dx) + y. The right side remains 0 since the derivative of a constant is 0. Setting x = 3 and y = -10 and solving for y' results in:
4(3) + 5(-10) + 5(-10)y' = 0
12 - 50 - 50y' = 0
-38 = 50y'
y' = -38/50
y' = -0.76
Therefore, y'(3) = -0.76.