Part 5/5:
For the expression ( y = \ln(\sin(x)) ), we again use the chain rule:
[
\frac{dy}{dx} = \frac{1}{\sin(x)} \cdot \frac{d}{dx}(\sin(x)) = \frac{\cos(x)}{\sin(x)} = \cot(x)
]
Conclusion
In summary, we have thoroughly explored the derivatives of logarithmic functions, emphasizing the concepts of implicit differentiation and the chain rule. The resulting derivatives for ( \log_a(x) ) and ( \ln(x) ) underline the utility of natural logarithms in calculus, as they lead to simpler forms. Understanding these functions is essential for anyone delving into calculus and its applications in various fields. Stay tuned for more mathematical insights and solutions!