Some Applications of the Chain Rule
Read this section to learn how to apply the Chain Rule. Work through practice problems 1-8.
Derivatives of Logarithms
Proof: We know that the natural logarithm is the logarithm with base , and for
We also know that , so using the Chain Rule we have Differentiating each side of the equation , we get that
The function is the composition of with , so by the Chain Rule,
Solution: (a) Using the pattern with , then
(b) Using the pattern with , we have .
We can use the Change of Base Formula from algebra to rewrite any logarithm as a natural logarithm, and then we can differentiate the resulting natural logarithm.
Change of Base Formula for logarithms: for all positive and .
Example 2: Use the Change of Base formula and your calculator to find and .
Practice 1: Find the values of and .
Putting in the Change of Base Formula, , so any logarithm can be written as a natural logarithm divided by a constant. Then any logarithm is easy to differentiate.
The second differentiation formula follows from the Chain Rule.
The number might seem like an "unnatural" base for a natural logarithm, but of all the logarithms to different bases, the logarithm with base e has the nicest and easiest derivative. The natural logarithm is even related to the distribution of prime numbers. In 1896, the mathematicians Hadamard and Valle-Poussin proved the following conjecture of Gauss: (The Prime Number Theorem) For large values of number of primes less than .