We will now obtain derivative formulas for the logarithmic and exponential functions and discuss the general relationships between and derivative of a one by one function and its inverse.
Natural logarithm plays a special role in calculus that can be motivated by differentiating where b is an arbitrary base. For this proposal, we will admit what is differentiable, and therefore continuous for x> 0. We will also need the limit
Using the definition of derivative, we obtain (with x rather than v as a variable).
But from the formula, we have = 1 / 1n B; so we can rewrite this derivative formula as
In the special case where b = e, we have = 1n and = 1, so this formula becomes
So, among all possible bases, the base b = e produces the simplest derivative formula for . This is one of the reasons why the natural logarithm function is preferred over all logarithms in the calculation.
When possible logarithm properties should be used to convert products, quotients and exponents into sums, differences and multiples of constants before differentiating a function involving logarithms.
Next: Logarithmic Differentiation