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Understanding how to differentiate a function that is the quotient of two other functions is a fundamental skill in calculus. The quotient rule is a derivative rule that allows us to find the derivative of a quotient of functions. It states, in simple terms, that the derivative of a function represented as the ratio \(\frac{u}{v}\) is given by the formula:

\[\frac{d}{dx}\left(\frac{u}{v}\right) = \frac{v\frac{du}{dx} - u\frac{dv}{dx}}{v^2}\]

In this formula, \(\frac{du}{dx}\) and \(\frac{dv}{dx}\) represent the derivatives of the numerator function \(u\) and the denominator function \(v\), respectively. The quotient rule essentially tells us to differentiate the top and bottom separately, multiply them by the opposite function, subtract the two products, and then divide by the square of the denominator.

Trigonometric functions are prevalent in calculus problems, and understanding their derivatives is crucial. The sine and cosine functions, denoted as \(\sin x\) and \(\cos x\), have straightforward derivatives, which are:

\[\frac{d}{dx}(\sin x) = \cos x\]
and
\[\frac{d}{dx}(\cos x) = -\sin x\]

These basic derivatives are often used as the building blocks for more complex differentiation problems, including those involving the quotient rule, as demonstrated in our exercise.

Once we have obtained a derivative expression through rules such as the quotient rule, simplification is often necessary to reach the final, most elegant form of the derivative. Simplification may involve combining like terms, factoring, canceling common factors, and applying trigonometric identities.

For instance, recognizing that \(\sin^2x + \cos^2x = 1\) can be a powerful tool in simplifying expressions. In simplification, always be on the lookout for opportunities to reduce the expression to its simplest form, making it easier to understand and work with.

Applying differentiation rules such as the product rule, quotient rule, chain rule, etc., can seem daunting at first. However, with practice, these rules become an intuitive part of solving calculus problems. When faced with a function, identify which rules are applicable, apply them methodically, and then simplify. Remember, the correct application of differentiation rules is a systematic process: identify the functions involved and their derivatives, apply the rules step by step, and simplify the resulting expression.

In the example problem, we applied the quotient rule expertly, differentiated the numerator and denominator as separate functions, and then simplified. By following these steps, differentiation becomes a much more approachable task.