Derivatives of Ln Functions

Introduction to derivative of ln function:

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.

Source Wikipedia.

Derivative Formula for Ln Functions:

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

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

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Derivative Ln Function Problems:

Derivative ln function problem 1:

Find the Derivative of given ln function : f(x) =  ln(15x - 12) with respect to x.

Solution:

Given ln function is   ln(15x - 12)

Let  u = 15x - 12                                           So, f(x) = ln u

`(du)/(dx)` = 15

f(x) = ln u

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

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

= `1/(15x - 12) (15)`

= `15/(15x - 12)`

Answer: The derivative of ln(15x - 12)  is `15/(15x - 12)`

Derivative ln function problem 2:

Find the Derivative of given ln function :  ln t17  with respect to t.

Solution:

Given ln function is   ln t17

Let z = ln t17                                                                                    we know, log an = n log a

So we can write the question as

z = ln t17 = 17 ln t

The derivative will be simply 17 times the derivative of ln t.

So the derivative of ln t17 is:

`d/dt` ( ln t17)  = 17 ` d/dt` (ln t)

= 17 `(1/t)`

= `17/t`

Answer:    The derivative of ln t17  is `17/t`

Derivative ln function problem 3:

Find the Derivative of given ln function :  z log6 12z.

Solution:

Given ln function is z log6 12z.

Let f(x) = z log612z

Now we use the Multiplication rule,

f'(x) = log612z + z (log612z)'

We know log ba =  `(log a) / (log b)`

` log_6 12z = ` `(log 12z ) / (log 6)`

So, f'(x) = log612z + z (log612z)'

= log612z + z ` 12/(12z log 6)`

= log612z + ` 1/log 6`

Answer:   The derivative of y log612z. is   log612z +` 1/log 6`

Derivative ln function problem 4:

Find the Derivative of given ln function : f(x) =  ln(x + y) with respect to x.

Solution:

Given ln function is  ln(x + y)

Let  u = x + y                                   ( y - constant)

`(du)/(dx)` = 1                                    So, f(x) = ln u

f(x) = ln u

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

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

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

= `1/(x + y)`

Answer: The derivative of ln(x + y)  is  `1/(x + y)`