Final answer:
To find dy/dx in terms of x and y, we use implicit differentiation. Applying the product rule and power rule, we differentiate both sides of the equation x^3y - x - 8y - 15 = 0 with respect to x. The resulting derivative equation is dy/dx = (8 - 3x^2y) / (x^3 - 8).
Step-by-step explanation:
To find dy/dx in terms of x and y, we will use implicit differentiation. Given the equation x^3y - x - 8y - 15 = 0, we will differentiate both sides with respect to x.
For the left side, we will use the product rule and the power rule. The derivative of x^3y with respect to x is 3x^2y + x^3(dy/dx). The derivative of -8y is -8(dy/dx). And the derivative of -15 with with respect to x is 0.
So, our derivative equation is: 3x^2y + x^3(dy/dx) - 8(dy/dx) = 0. Solving this equation for dy/dx gives us: dy/dx = (8 - 3x^2y) / (x^3 - 8).