The statement that is equivalent to f' (1) given f(x) is continuous for all real x is (B) {eq}f(1+h)-f(1) \over h {/eq} where h is not equal to zero.What is the meaning of f' (1)?The derivative of the function f(x) evaluated at x = 1 is represented by f'(1). If f'(1) exists, it implies that the slope of the tangent line at x = 1 exists. In other words, the slope of the function f(x) at x = 1 is equivalent to f'(1).If we are given that f(x) is continuous for all real numbers, we can find an equivalent expression to f'(1). A possible answer is:{eq}f(1+h)-f(1) \over h {/eq}The expression {eq}f(1+h)-f(1) \over h {/eq} is a mathematical definition of the slope of the line joining two points of the curve f(x) (x, f(x)) and (1+h, f(1+h)). Note that the slope of the line is calculated using the formula (rise/run), where the rise is equal to f(1+h)-f(1) and the run is h. Therefore, this expression is an equivalent representation of f'(1).For small values of h, the slope of the secant line joining (1, f(1)) and (1 + h, f(1 + h)) approximates the slope of the tangent line to the graph at x = 1. Therefore, the limit of {eq}{f(1+h)-f(1) \over h}{/eq} as h approaches zero is equivalent to f'(1).