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) = 2x

  2. f(x) = 3sin x  

  3. f(x) = xx   


Solution

  1. We write

            2x = ex ln 2

    Now use the chain rule

            f '(x) = (ex ln 2)(ln 2) = 2x ln 2

  2. We Write

            3sin x = e(sin x)(ln 3)

    Now use the chain rule

            f '(x) = (e(sin x)(ln 3))(cos x)(ln 3) 

            = (3sin x) (cos x) (ln 3)

  3. We Write

            xx = ex ln x

    Notice that the product rule gives

            (x ln x)' = 1 + ln x

    So using the chain rule we get

            f '(x) = ex ln x (1 + ln x) = xx (1 + ln x)


Exercises

Find the derivatives of 

 

  1.  x2x + 1    x^(2x + 1)  [ (2x + 1) / x  + 2 ln x ]

  2. x4          4x^3


 

Logs With Other Bases

    

         

Definition  

                        ln x
     loga =             
                       
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) = 2x  

Solution

        2x dxex ln 2 dx         u = x ln 2,    du = ln 2dx

                      1
            =               eudu 
                    ln 2

                 1                               ex ln 2     
        =                 eu  +  C  =                 
                ln 2                             ln 2

                 2x                       
        =                +  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 = Pert  


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?

    2 points per month

 



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