Final answer:
The slope of the graph for the given circle equation is found using implicit differentiation and is given by the negative ratio of (x - 3) to (y - 4), resulting in the slope - (x - 3) / (y - 4) at any point (x, y).
Step-by-step explanation:
The slope of the graph at any point (x, y) for the equation (x - 3)² + (y - 4)² = 25 can be determined by finding the derivative of the function implicitly because the equation describes a circle, not a line. First, we use the chain rule to differentiate both sides of the equation with respect to x. The derivative of (x - 3)² with respect to x is 2(x - 3), and the derivative of (y - 4)² with respect to x is 2(y - 4)dy/dx considering y as a function of x. Setting the derivative of the constant 25 equal to zero, the expression for the slope, dy/dx, at any point (x, y) would be the negative ratio of the derivative of (x - 3)² to the derivative of (y - 4)², which simplifies to - (x - 3) / (y - 4) after factoring out the 2s and moving terms around.