Introduction to what is the derivative of ln

The log representation is given for a logarithm with its base as 10. “ln” is the representation of logarithm with the base as “e”. Derivative on log cannot be done unless it has the base as “e”. 

Let us learn about what is the derivative of ln.

 

So, what is the derivative of ln?

Let y = ln(x).

Therefore the derivative of y is `dy/dx` = `(d(ln(x)))/dx`

                                                             = `(1)/(x)` .

 So, it is clear that when we differentiate the “ln” function, it will be `(1)/(x)`, where x is the function inside the “ln”.

 

Let us see more on what is the derivative of ln

 

Before knowing more about the derivative of ln, we need to know some basic operations on ln.

(i) ln(ab) = ln(a) + ln(b)

(ii) ln(`a/b` ) = ln(a) - ln(b)

(iii) ln(an) = n ln(a)

(iv) logb (a) = ln (a)/ ln (b)

(v)  ln(ab)n = n (ln(a) + ln(b))

(vi) ln(`a/b` )n = n (ln(a) - ln(b))

So, to apply the formula for the differentiation of “ln”, we need to have the clear idea about the operation with ln. It is as same as log as well.

Ex 1: What is the derivative of y = ln(x2)?

Sol:  `dy/dx` = `(d(lnx^2))/(dx)` = 2 `(d(lnx))/(dx)`

                              = 2 (`(1)/(x)`)

                              = `(2)/(x)`.

Ex 2: What is the derivative of y = ln(1 + x)?

Sol:  `dy/dx` = `(d(ln(1+x)))/(dx)`

                      = `[(1)/(1+x)]`.                            

 

More examples on what is the derivative of ln

 

Ex 1: What is the derivative of  y = ln[(1 + x)(2+x)] ?

Sol:    `dy/dx` = `(d(ln(1+x)(2+x)))/(dx)`  

                   =  `(d(ln(1+x)))/(dx)` `(d(ln(2+x)))/(dx)`

                   = `(1)/(1+x)` + `(1)/(2+x)`

 

Ex 2: What is the derivative of y = ln[(1 + x)/(2+x)]?

Sol:  `dy/dx` `(d(ln((1+x)/(2+x))))/(dx)` <br>

                  =  `(d(ln(1+x)))/(dx)` `(d(ln(2+x)))/(dx)`

                  = `(1)/(1+x)` - `(1)/(2+x)`

 

Ex 3: What is the derivative of  y = ln[ex] ?

Sol: `dy/dx` =`(d(ln(e^x)))/dx`   

                   =  `(d(x))/dx`                      [Here, since the base of ln is e, ln(ex) = x]

                   = 1.

I hope this basic discussion on “what is the derivative of ln” would help you to get into the logarithmic differentiation further.