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The chain rule is an essential tool in calculus for finding the derivative of composite functions. When a function is composed of one function inside another, the chain rule allows us to differentiate it step by step. For example, if a function u can be expressed as a composition of two functions, say u=g(f(x)), then the derivative of u with respect to x would be found using the chain rule as:
\[\frac{du}{dx} = \frac{dg}{df} \cdot \frac{df}{dx}\]
This becomes invaluable when dealing with more complex functions, like logarithmic functions involving trigonometric expressions. It helps to break down the derivative process into manageable parts, which makes it easier to tackle even the most intimidating of functions.

Understanding the properties of logarithms is crucial when differentiating logarithmic functions. A logarithm, at its core, is an exponent. The two most pertinent properties that often come into play when differentiating are:

• The product property: \(\log(a\cdot b) = \log(a) + \log(b)\).
• The power property: \(\log(a^b) = b \cdot \log(a)\).

In the exercise, we used the power property to simplify the square root expression into a more 'derivative-friendly' form. Recognizing the right property to use and applying it correctly simplifies the function and sets the stage for an easier differentiation process.

In calculus, derivatives represent the rate at which a function is changing at any given point, and it is one of the fundamental concepts for understanding change and motion. For our function involving a logarithm, the derivative tells us how quickly the logarithmic function is changing in response to changes in x. A solid grasp of how to take these derivatives is necessary not just for solving textbook problems but for understanding a wide array of scientific and engineering problems as well.

Trigonometric functions—such as sine, cosine, and tangent—are functions that relate the angles of a triangle to its side lengths. They are periodic and continuous, making them intricate to both theoretical and applied mathematics, including calculus. When dealing with derivatives of these functions, one must be familiar with their basic derivatives:

• The derivative of \(\sin(x)\) is \(\cos(x)\).
• The derivative of \(\cos(x)\) is \(-\sin(x)\).

In our exercise, knowing that the derivative of \(\cos(x)\) is \(-\sin(x)\) is vital for applying the chain rule correctly. Trigonometric functions often appear in the study of waves, oscillations, and many other natural phenomena, so their derivatives are widely used across different fields of science and technology.