2.4.4 Derivatives of More General Exponential
         Functions

Have we really accomplished something by introducing \(e\) and concentrating on the one function exp ( t ) = e t? Suppose we consider functions u ( t ) = e 3 ⁢ t or v ( t ) = e 5 ⁢ t or w ( t ) = e 2.35 ⁢ t. How do we differentiate these functions? In general, we need to know how to differentiate functions of the form e k ⁢ t, given that the derivative of e t is e t.

Activity 6  

1. Use your graphing tool to plot f ( t ) = e k ⁢ t, g ( t ) = f ( t + 0.001 ) - f ( t ) 0.001, and their ratio, g ( t ) f ( t ), for \(k=2, 3, 4.\)
   
   
2. Conjecture a general formula for d d t e k ⁢ t.

Comment on Activity 6

Checkpoint 4

Now we establish a general formula for differentiation of exponential functions. Suppose \(A\) and \(k\) are constants. Then

Conclusion:

d d t A ⁢ e k ⁢ t = k ⁢ A ⁢ e k ⁢ t.

Example 1   Calculate the derivative of 5 ⁢ e 3 ⁢ t.

Solution    d d t 5 ⁢ e 3 ⁢ t = 3 ⋅ 5 ⁢ e 3 ⁢ t = 15 ⁢ e 3 ⁢ t