QUESTION 1 1.1 Differentiate \( y=\log \sqrt{3 x} \) by using a function of a function (chain rule).

To differentiate \( y=\log \sqrt{3 x} \), we can rewrite it as \( y=\log (3 x)^{1/2} \) or \( y=\frac{1}{2} \log (3 x) \). Using the chain rule, we differentiate as follows: First, differentiate \( y \) with respect to \( (3 x) \): \[ \frac{dy}{d(3x)} = \frac{1}{(3x)} \] Then, multiply by the derivative of the inner function \( 3x \): \[ \frac{dy}{dx} = \frac{1}{2} \cdot \frac{1}{(3x)} \cdot 3 = \frac{3}{6x} = \frac{1}{2x} \] So, the derivative is \( \frac{1}{2x} \). Now let’s take a fun detour! Did you know? Logarithms were first introduced by John Napier in the early 17th century as a tool to simplify complex calculations, especially in astronomy. His work laid the foundation for modern logarithmic tables, making calculations much more manageable for scientists. If you're looking to apply this knowledge practically, understanding the logarithmic functions in settings like data analysis is crucial. For instance, when dealing with large datasets, logarithmic transformations can help stabilize variance and allow for better model fitting! This is particularly handy when working with exponential growth patterns commonly seen in various real-world scenarios, such as population growth, finance, and even social media trends!