Derivatives of logarithmic functions

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).

Like this,

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.

Example 1


Solution. From

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.

Example 2

Next: Logarithmic Differentiation