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Home Exponential and Logartihmic Functions Derivatives of Exponential Functions and the Number e Examples Example 3
 
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Example 3

Find d(ax)/dx. We use the formula

08_exp-log_functions-111.gif

Put u = x loge a. Then ax = ea, so

08_exp-log_functions-112.gif

This example shows that the derivative of ax is equal to the constant loge a times ax itself. Figure 8.3.3 shows the graph of y = ax for various values of a > 0.

08_exp-log_functions-114.gif

Figure 8.3.3

 

The slope of the curve y = ax at x = 0 is always equal to loge a. For all values of a > 0, ax is positive for all x, so the derivative has the same sign as logea. The three possibilities are shown:

a > 1

loge a > 0

ax increasing for all x

a = 1

loge a = 0

ax = 1 for all x

0 < a < 1

loge a < 0

ax decreasing for all x

Home Exponential and Logartihmic Functions Derivatives of Exponential Functions and the Number e Examples Example 3

Last Update: 2006-11-15