Final answer:
To find dy/dx at (-8,1), differentiate the given equation with respect to x. Substitute the values of x and y to find the slope at (-8,1).
Step-by-step explanation:
To find dy/dx at (-8,1), we need to differentiate the given equation with respect to x. First, rearrange the equation to isolate y: xy = 40y/x + 3. Multiplying both sides by x gives x^2 * y = 40y + 3x. Next, differentiate both sides with respect to x using the product rule. Differentiating x^2 * y gives 2xy + x^2 * dy/dx, and differentiating 40y gives 40 * dy/dx. Finally, rearrange the equation to solve for dy/dx: 2xy + x^2 * dy/dx = 40 * dy/dx + 3x. Subtract 40 * dy/dx from both sides and isolate dy/dx by factoring out the common term: dy/dx * (x^2 - 40) = 2xy - 3x. Divide both sides by (x^2 - 40) to obtain dy/dx = (2xy - 3x) / (x^2 - 40). Substitute the values x = -8 and y = 1 into the equation to find dy/dx at (-8,1).