Derivative of Exponential Function

Motivation:

 

                          After knowing the derivative of logarithm function and the general way to find the derivative of inverse function,  it is natural to take a look at exponential function.  Usually, exponential functions are with the type  y= ca^bx .  And it is useful in the application of finding the return of investment of several years, or the study of the half-life of radioactive materials.  Since we have chain rule to deal with the variant forms of the function, we can just put our focus on the function of the simple form:

 

                                                          y=f(x)=a^x

 

                           Let’s start with the basic definition of derivative function:

                                

                                   f(x+h)-f(x) = a^x+h - a^x

                                                     = a^x( a^h – 1) 

     

                           It does not seem that the problem can be tackled via this straightforward way.  Thus,  we need to translate it into the form we are familiar with.  Naturally, we will think about logarithm function because the way we define logarithm is from finding the solution for the exponential equation.

 

 

 

 

 

Theorem:  y=f(x)=a^x ,  where a>0 .    Then

                              

                                                 

       Proof:

                                        y=a^x

                                       ln(y) = ln(a^x) = x ln(a)

                                 = ln(a)

                                 = y ln(a) = a^x ln(a) 

 

                               Actually, this is exactly the way we use to find the derivative of inverse function.  The idea behind it is “chain rule” .  Using chain rule directly is easier than memorizing the previous result.    If  a=e , where e is the base of natural logarithm, the result is simpler:

                                                          f(x)=e^x

                                                       f’(x) = e^x

 

                     Its form does not change after differentiation.  Usually, people use this property to find the solution for some differential equations.  For example,

                                                 

                                                             =y

 

                      Naturally, the answer is y(x)=e^x + C , where C is a constant.

 

 

Example:  f(x)=e^2x + e^3x ,  where e is the base of natural logarithm. Find f’(x) and f”(x).

 

         Sol:

                        By chain rule,

                                      f’(x)=2e^2x + 3e^3x 

                                      f”(x)=4e^2x + 9e^3x

 

                        And it is interesting to know

 

                                         f”(x) – 5f’(x)+6f(x) = 0.