We use logarithms to help us differentiate.

Logarithms were originally developed as a computational tool. The key fact that made this possible is that:

Before the days of calculators and computers, this was critical knowledge for anyone in a computational discipline.

The moral is:

Logarithms allow us to use addition in place of multiplication.

Logarithmic differentiation

When taking derivatives, both the product rule and the quotient rule can be cumbersome to use. Logarithms will save the day. A key point is the following which follows from the chain rule. Let’s look at an illustrative example to see how this is actually used.

The process above is called logarithmic differentiation. Logarithmic differentiation allows us to compute new derivatives too.

The next example will be useful when we want to use logarithmic differentiations for functions that assume negative values.

Proof of the power rule

Finally, recall that previously we only proved the power rule for positive integer exponents. Now we’ll use logarithmic differentiation to give a proof for all real-valued exponents. We restate the power rule for convenience sake:

While logarithmic differentiation might seem strange and new at first, with a little practice it will seem much more natural to you.