Part 4/5:
An essential property worth noting is that while both ( \log_a(x) ) and ( \ln(x) ) can be transformed into each other through the change of base formula, natural logarithms are frequently preferred in real-world applications. This is due to their easier differentiation and integration properties, particularly because ( e ) is a transcendental number with unique properties in calculus.
Examples of Derivatives
Let’s explore a couple of specific examples to clarify further how these derivatives are applied.
Example 1: Derivative of ( y = \ln(x^3 + 1) )
To differentiate ( y = \ln(x^3 + 1) ):
First, apply the chain rule:
[
\frac{dy}{dx} = \frac{1}{x^3 + 1} \cdot \frac{d}{dx}(x^3 + 1) = \frac{3x^2}{x^3 + 1}
]