Other Bases

Exponentials With Other Bases

              

Definition   Let a > 0 then           ax = ex ln a

Examples

Find the derivative of the following functions

1. f(x) = 2^x
   

2. f(x) = 3^{sin x}  
   

3. f(x) = x^x   
   
   
   

Solution

1. We write
   
           2^x = e^{x ln 2}
   
   Now use the chain rule
   
           f '(x) = (e^{x ln 2})(ln 2) = 2^x ln 2
   

2. We Write
   
           3^{sin x} = e^{(sin x)(ln 3)}
   
   Now use the chain rule
   
           f '(x) = (e^{(sin x)(ln 3)})(cos x)(ln 3) 
   
           = (3^{sin x}) (cos x) (ln 3)
   
   

3. We Write
   
           x^x = e^{x ln x}
   
   Notice that the product rule gives
   
           (x ln x)' = 1 + ln x
   
   So using the chain rule we get
   
           f '(x) = e^{x ln x} (1 + ln x) = x^x (1 + ln x)
   
   

   ---

Exercises

Find the derivatives of 

1.  x^{2x + 1}    
   

2. x⁴         
   

Logs With Other Bases

         

Definition                           ln x      loga x  =                                      ln a

    

Examples

Find the derivative of the following functions

1. f(x) = log₄ x 
   

2. f(x) = log (3x + 4)
   

3. f(x) = x log(2x) 
   

Solution

1. We use the formula
   
           f(x) = ln x / ln 4
   
   so that 
   
           f '(x) = 1/(x ln 4)
   

2. We again use the formula
   
                           ln (3x + 4)
           f (u)  =                       
                              ln 10
   
   
   now use the chain rule to get
   
                            3 ln (3x + 4)
           f '(x)  =                           
                                 ln 10
   
   
   

3. Use the product rule to get
   
           f '(x) = log(2x) + x(log 2x)'
   
   Now use the formula to get
   
                              ln 2x          x ( ln 2x)'
           f ' (2x)  =                   +                 
                              ln 10             ln 10
   
   
   
   The chain rule gives
   
                                ln 2x                  2x
           f ' (x)  =                        +                  
                                ln 10               2x ln 10
   
   
                                 ln 2x                 1
           f ' (x)  =                        +               
                                ln 10               ln 10
   
   

   ---

    

Integration

Example

   Find the integral of the following function

        f(x) = 2^x  

Solution

        2^x dx =  e^{x ln 2} dx         u = x ln 2,    du = ln 2dx

                      1
            =               e^udu 
                    ln 2

                 1                               e^{x ln 2}     
        =                 e^u  +  C  =                 
                ln 2                             ln 2

                 2^x                       
        =                +  C  
                ln 2                    

Application:  Compound Interest

Recall that the interest formula is given by:

            A = P(1 + r/n)ⁿ 

where n is the number of total compounds before we take the money out, r is the interest rate,
P is the Principal and A is the amount the account is worth at the end.  
If we consider continuous compounding, we take the limit as n approaches infinity we arrive at

            A = Pe^rt  

Exercise

Students are given an exam and retake the exam later.  The average score on the exam is

            S = 80 - 14ln(t + 1) 

where t is the number of months after the exam that the student retook the exam.  At what rate is the average student forgetting the information after 6 months?

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