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**

Find

** 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**