Study about derivative of logarithmic function

Introduction :

In calculus, the derivative is a measure of how a function changes as its input changes.  The process of finding a derivative is called differentiation. 

 The derivative of logarithmic function is the derivatives of inverse ofexponential function since the inverse of exponential function is logarithmic function. To study about derivative of logarithmic function, we must know the derivative identities for logarithmic function as follows 

`d/(dx)` (ln u) = `1/u` `(du)/(dx)`

`d/(dx)` (logx u) = `1/((ln a)u)` `(du)/(dx)`

In this article, we are going to study about derivative of logarithmic function with example and practice problems.

               (Source: Wikipedia)


Study about derivative of logarithmic function with example problems:


Example 1:

Find the derivative of y = 3ln (7x)

Solution:

  Step 1: Given function

  y = 3ln (7x)

  Using logarithmic identities, the given function can be written as

  y = 3ln 7 + 3ln x

  Step 2: Differentiate the function y = 3ln 7 + 3ln x with respect to ' x '

  `(dy)/(dx)` = 3(0) + 3 `1/x`

  = `3/x`


Example 2:

Find the derivative of y = 4ln sin x

Solution:

  Step 1: Given function

  y = 4ln sin x

  Step 2: Differentiate the function y = 4ln sin x with respect to ' x '

  `(dy)/(dx)` = 4 `(1/(sin x))` `d/(dx)` sin x

  = `4/(sin x)` cos x


Example 3:

Find the derivative of y = 2ln (7x + 9)

Solution:

  Step 1: Given function

  y = 2ln (7x + 9)

  Step 2: Differentiate the function y = 2ln (7x + 9) with respect to ' x '

  `(dy)/(dx)` = 2 `(1/(7x + 9))` `d/(dx)` (7x + 9)

  = `2/(7x + 9)` (7)

  = `14/(7x + 9)`


Study about derivative of logarithmic function with practice problems:


1) Find the derivative of the function y = 4ln (9x)

2) Find the second derivative for the function y = 4ln tan x

3) Find the derivative for the function y = 4ln (7x + 3)


Solutions:

1) `4/x`

2) `4/(tan x)` sec2 x

3) `28/(7x + 3)`