Logs and Derivatives

Logs and Derivatives

Definition of the Natural Logarithm

Recall that

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What is

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Definition:  For x > 0 we define

Note:  The Second Fundamental Theorem of Calculus tells us that 

        d/dx
(ln x) = 1/x

Properties of ln x

1. ln 1 = 0

2. ln(ab) = ln a + ln b
   

3. ln(an) = n
   ln a
   

4. ln(a/b) = ln a –
   ln b

Proof of (3)  

So that  

        ln(xn) 

and 

        n ln x 

have the same derivative.  Hence

        ln(xn) =
n ln x + C

Plugging in x = 1 we have that C =
0.

Definition of e

Let e be such that 

        ln e = 1 

ie. 

Examples and Exercises

Example

Find the derivative of 

        ln (x2  + 1)

Solution

We use the chain rule with 

        y = ln u,   
u = x2 + 1 

                                    
  2x
        y‘  =  (2x)(1/u) 
=             
   
                                  
  x2 + 1

Find the derivatives of the following functions:

1. ln (lnx)       

2. (ln x)/x        
   

    

3. (ln x)2       
   

4. ln (sec x)       
   

5. ln (csc x)       
   

6. Show that 

           3 ln x – 4 

   is a solution of the differential equation

           xy” + y‘ = 0

7. Find the relative extrema of 

           x
   ln x       

8. Find the equation of the tangent line to 

           y = 3x2 –
   ln x 

   at (1,3)       

9. Find dy/dx for

           ln(xy) +
   2x2 = 30       

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