1. John Napier
1550 to 1617
John Napier was a Scottish scholar who is best known for his
invention of logarithms, but other mathematical contributions
include a mnemonic for formulas used in solving spherical triangles
and two formulas known as Napier's analogies.

3. We would like to find its derivative.

4. Using the chain rule we get
Example

5. Example

6. Hence, the generalized derivatives for exponential functions:

7. Now we want to find the derivative of logarithmic functions.
The generalized derivative is:

8. Example

9. Example

11. To find the derivative of any logarithmic function of any base, you can use the change of base rule for logs:
Change of Base to e
The generalized derivative formula of the log of any base is:

12. To expand the domain of logarithmic functions we take the absolute
value of the argument. So lets examine
It can shown that,
The derivative is the same!!

13. Example

14. Example

16. Example 2)

17. Example

18. So in conclusion, the generalized derivatives of exponential and
logarithmic functions are: