Final answer:
To find ²y/x² at a specific point using implicit differentiation, differentiate both sides of the equation with respect to x, substitute the given point into the equation, and solve for dy/dx. At the point (0,3), ²y/x² is equal to 0.
Step-by-step explanation:
To find ²y/x² at a specific point using implicit differentiation, we will start by differentiating both sides of the equation with respect to x. The derivative of 9x² with respect to x is 18x, and the derivative of y² with respect to x is 2y * (dy/dx). Therefore, the derivative of the left side of the equation is 18x + 2y * (dy/dx). The derivative of the right side of the equation is 0, since it is a constant. Now we can substitute the given point (0,3) into the equation to find the value of dy/dx.
Substituting x=0 and y=3 into the derivative equation, we have 18(0) + 2(3) * (dy/dx) = 0. Simplifying this equation gives us 6(dy/dx) = 0, which implies that dy/dx = 0/6 = 0. Therefore, ²y/x² at the point (0,3) is equal to 0.