Derivatives of Exponential Functions In this slide show, we will learn how to differentiate such functions. This will enable us to solve many problems which occur naturally: This will enable us to solve many problems which occur naturally: Such as: Such as: growth and decay Money invested at compound interest The decay of radioactive material AND AND Growth of bacteria cultures in medicine. Question: Why cant we use the Power Rule for derivatives to differentiate y = b x ? Question: Why cant we use the Power Rule for derivatives to differentiate y = b x ? ANSWER: The Power Rule is valid only when the function has the variable (usually x) in the base and a constant for the exponent, and not, the other way around. FIRST Recall: FIRST Recall: (1) (1) And recall as well: And recall as well: (2) (2) (property of limits which says the factor c is constant as x a, and does not depend on x. For example: For example: (1) (1) OR Since the factor 6 is constant and does not depend on x (2) (2) Or Since the factor x 3 is constant and does not depend on h (where did that come from??) So we must try to find the derivative of the exponential function So we must try to find the derivative of the exponential function From first principals, that is using the formula: Thus, with y = 2 x, we get Thus, with y = 2 x, we get And so, And so, Which we recognize from earlier And so The question now is what is the value of The question now is what is the value of Or more importantly, the value of the limit; Since the factor 2 x is constant and does not depend on h One way evaluate this limit is to One way evaluate this limit is to graph the function graph the function On our graphing calculator and see what the value of y is as h is approaching 0 (i.e the y intecept). (before we go on, can you think of another way? By going to table set and setting Tbl=-.5 and Tbl =.1 We will be able to see the value of y near where x is 0. We can see from the table the values near x = 0. (why is there no value when x is zero?) If we average the two values around x = 0 we get approximately Thus, So getting back to our original problem, So getting back to our original problem, if y = 2 x, then if y = 2 x, then The conclusion is that if y = 2 x then its The conclusion is that if y = 2 x then its derivative, dy/dx is equal to a multiple derivative, dy/dx is equal to a multiple of itself. That is, of itself. That is, dy/dx = (0.693) 2 x dy/dx = (0.693) 2 x Let us look at changing the value of b from 2 to 2.6 If y = (2.6) x, then Once again, if we use our calculators Once again, if we use our calculators to find the value of this limit, to find the value of this limit, we discover that we discover that Therefore, Therefore, b Which leads us to the following derivatives: Which leads us to the following derivatives: Looking back at the table: Looking back at the table: b It may have occurred to you that there is a value of b between 2.7 and 2.8 for which = 1 Let us call this value of b, e. Let us call this value of b, e. And so we define e to be that value of And so we define e to be that value of b between 2.7 and 2.8 such that b between 2.7 and 2.8 such that =1. Or in other words, In which case, if y =e x, In which case, if y =e x, then dy/dx = e x then dy/dx = e x = e x What makes this interesting, What makes this interesting, and worth the wait, is that we have and worth the wait, is that we have a function, y = e x whose derivative, a function, y = e x whose derivative, dy/dx is itself!! dy/dx is itself!! This is a portrait of Euler, the discoverer of the concept of e. He was the original e-male