Introduction :

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 Formulas for ln:

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

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

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

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

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

Derivative problems:

Derivative problem 1:

Calculate the derivative of ln 5 with respect to x.

Solution:

Given ln function is ln 5

Let z = ln 5 we know, log an = n log

So the derivative of ln 5 with respect to x:

`d/(dx)` ( ln 5) = `d/(dx)` ( ln 5)

Here, no variable in terms of x. So it is a constant

Derivative of constant is 0

So, = `d/(dx)` ( ln 5)

= 0

Answer: The derivative of ln 5 is 0.(zero)

Derivative problem 2:

Calculate the Derivative of log 5u. with respect to u

Solution:

Given ln function is log 5u

Let f(x) = log 5u

Now we use the Multiplication rule, log (AB) = log A + log B

f(x) = log (5u) = log 5 + log u

So, f'(x) = `d/(du)` (log 5u) = `d/(du)` (log 5)+ `d/(du)` (log u)

= 0 + `1/u`

= `1/u`

Answer: The derivative of log 5u. is `1/u`

Derivative problem 3:

Find the derivative of ln y5 with respect to y.

Solution:

Let z = ln y5 we know, log an = n log a

So we can write the question as

z = ln y5 = 5 ln y

The differentiation will be simply 5 times the derivative of ln y.

So the derivative of ln y5 is:

`d/dy` ( ln y5) = 5 ` d/dy` (ln y)

= 5 `(1/y)`

= `5/y`

Answer: The derivative of ln y5 is `5/y`