please do the following problems and show the work

Derivatives of Exponential and Logarithmic Functions Remark 26 Recall from Definition 2 that the natural exponential base e is defined by the limit e := lim 1 + - . Notice that this is equivalent to the limit e := lim (1 + h) . 56. (a) Find the difference quotient for the function y = ed. Simplify as much as possible. (b) Explain why Remark 26 is true. (c) Use the observation in Remark 26 to show that lim = 1. h-+0 h (d) Use your results to fill in the conclusion for Theorem 27. Theorem 27 (Derivative of the Natural Exponential Function) d dr 57. Calculate the following derivatives: (a) d 2x + 1 dr d (b) dr d (C) - Op (e)58. Define the natural logarithm function In(r) to be the inverse function of y = e. Sketch the graph of y = ed and the graph of y = In(r) on the same set of axes. Give the domain and range of each function. Explain the relationship between the two functions and their graphs