Final answer:
To find the slope of the curve at the given point, differentiate the equation implicitly, substitute the values of x and y into the derivative equation, and solve for dy/dx. The slope of the curve at the point (2, 4) is -2.
Step-by-step explanation:
To find the slope of the curve at the given point, we need to differentiate the equation implicitly and then substitute the x and y values of the given point into the derivative equation.
Given equation: x^2y - 2x^2 - 8 = 0
Differentiating implicitly:
2xy + x^2(dy/dx) - 4x = 0
Now, substitute x = 2 and y = 4 into the differential equation:
2(2)(4) + (2^2)(dy/dx) - 4(2) = 0
16 + 4(dy/dx) - 8 = 0
4(dy/dx) = -8
dy/dx = -2
Therefore, the slope of the curve at the point (2, 4) is -2.