Slope of a Nonlinear Curve
Finding the slope of a straight line of the form Y = a + bX is rather simple: the slope is given by b. What about the more general case of the unspecified equation Y = f(X)—How do we find its slope? If f(X) is nonlinear, its slope may be different at each point, but as long as the function is continuous and differentiable, its slope is given by the derivative of f(X) evaluated at that point.
Consider for example the function Y = f(X) = X 2 + 4. Suppose we want to know its slope at the point X = 3, Y = 13. The derivative of this function is f ’(X) = 2X, which takes on the value 6 when X = 3. Hence, the slope of this equation is 6 at the point (3, 13).