Chapter 3 | Derivatives
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3.9 | Derivatives of Exponential and Logarithmic Functions Learning Objectives 3.9.1 Find the derivative of exponential functions. 3.9.2 Find the derivative of logarithmic functions. 3.9.3 Use logarithmic differentiation to determine the derivative of a function.
So far, we have learned how to differentiate a variety of functions, including trigonometric, inverse, and implicit functions. In this section, we explore derivatives of exponential and logarithmic functions. As we discussed in Introduction to Functions and Graphs , exponential functions play an important role in modeling population growth and the decay of radioactive materials. Logarithmic functions can help rescale large quantities and are particularly helpful for rewriting complicated expressions. Derivative of the Exponential Function Just as when we found the derivatives of other functions, we can find the derivatives of exponential and logarithmic functions using formulas. As we develop these formulas, we need to make certain basic assumptions. The proofs that these assumptions hold are beyond the scope of this course. First of all, we begin with the assumption that the function B ( x ) = b x , b >0, is defined for every real number and is continuous. In previous courses, the values of exponential functions for all rational numbers were defined—beginning with the definition of b n , where n is a positive integer—as the product of b multiplied by itself n times. Later, we defined b 0 =1, b − n = 1 b n , for a positive integer n , and b s / t = ( b t ) s for positive integers s and t . These definitions leave open the question of the value of b r where r is an arbitrary real number. By assuming the continuity of B ( x ) = b x , b >0, we may interpret b r as lim x → r b x where the values of x as we take the limit are rational. For example, we may view 4 π as the number satisfying
4 3 <4 π <4 4 , 4 3.1 <4 π <4 3.2 , 4 3.14 <4 π <4 3.15 , 4 3.141 <4 π <4 3.142 , 4 3.1415 <4 π <4 3.1416 ,….
As we see in the following table, 4 π ≈77.88.
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