Introduction to derivative of ln y:

In mathematics, specifically in calculus and complex analysis, the logarithmic derivative of a function f is defined by the formula Where  f ′ is the derivative of f. `f'/f`. When f is a function f(x) of a real variable x, and takes real, strictly positive values, this is equal to the derivative of ln (f); or, the derivative of the natural logarithm of f. This follows directly from the chain rule.





Derivative ln formulas:


1. `d/dx` (ln x) = `1/x`

2. `d/dx` (logb u) = `1/(u ln b)` `(du)/(dx)`

3. `d/dx` (logb x) = `1/(x ln b)`

4.` d/dx`ln f(x) = `(f'(x))/(f(x))`

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

Derivative of ln y:

            Derivative of  ln y , with respect to y.

                           Let    z = ln y

                                 ez  =  y

     Differentiate on both sides with respect to y. we have to multiply by `dz/dy` on the left hand side.

                                ez  =  y

                       ez `dz/dy`   =  1

    Divided by ez on both sides, So we get

                           `dz/dy`   = ` 1/e^z`

           And then, Substitute ez = y, So we get

                           `dz/dy`   =  `1/y`

         The Derivative of ln y = `1/y`


Derivative problems:


Derivative problem 1:

          Determine the Derivative of f(t) =  ln(16t - 4) with respect to t.


                               Let  u = 16t - 4         So, f(t) = ln u

                                      `(du)/(dt)` = 16

                                      f(t) = ln u

                              `d/dt` (f(t))  = `d/dt` ( ln u)                                                  we know,  `d/dx` (ln u) = `1/u` `(du)/(dx)`   

                                           = `1/u `` (du)/(dt)`

                                           = `1/(16t - 4) (16)`

                                           = `16/(16t - 4)`

        Answer: The derivative of ln(16t - 4)  is `16/(16t - 4)`

Derivative problem 2:      

Determine the derivative of  ln y12  with respect to y.


                                       Let z = ln y12                                                                                    we know, log an = n log a

                  So we can write the question as

                                         z = ln y12 = 12 ln y

              The derivative will be simply 12 times the derivative of ln y.

                              So the derivative of ln y12 is:

                                  `d/dy` ( ln y12)  = 12 ` d/dy` (ln y)

                                                      = 12 `(1/y)`

                                                      = `12/y`

Answer:    The derivative of ln y12  is `12/y`

Derivative problem 3:     

Calculate the first derivative of function  log4 y


                           Let f(x) = log4y

               we know logarithmic relationship log nm =  `(log m) / (log n)`

                                f(x) = log4y = `(log y) / (log 4)`

                             Derivative of log y =` 1/y`

                                            So, f'(x) = `1/(y log 4)`

           Answer:  The derivative of log4y is `1/(y log 4)` 

Derivative problem 4:       

        Determine the derivative of function  y log 6y.


                         Let f(x) = y log 6y

                    Now we use the Multiplication rule,

                               f'(x) = log 6y + y (log 6y)'

                               We know log ba =  `(log a) / (log b)`

                                        ` log 6y = ` `(log 6y ) / (log 10)`     

                              So, f'(x) = log 6y + y (log 6y)'

                                          = log 6y + y ` 6/(6y log 10)`

                                          = log 6y +` 1/log 10`

 Answer:   The derivative of y log 6y. is   log 6y +` 1/log 10`


Derivative practice problems:

Derivative practice problem 1:

   Find the first derivative of f(z) = 5z with respect to z.

       Answer: 5z ln 5

Derivative practice problem 2:

       Find the first derivative of ln 14y with respect to y.

       Answer:     `1/y`