91Ó°ÊÓ

When dealing with derivatives of functions that are expressed as a ratio of two different functions, the Quotient Rule becomes an essential tool. It provides a systematic approach for differentiating such functions. In mathematical terms, if we have two functions, \( f(x) \) and \( g(x) \), where \( g(x) eq 0 \), the derivative of their ratio is given by:

• \( \frac{d}{dx}\left( \frac{f(x)}{g(x)} \right) = \frac{f'(x)g(x) - f(x)g'(x)}{(g(x))^2} \)

This formula helps in isolating the effect of each function's rate of change upon the whole fraction. In our specific problem, \( f(x) = \sin x \) and \( g(x) = \cos 2x \). This means that we first find the derivatives: \( f'(x) = \cos x \) and, applying the Chain Rule to \( g(x) \), \( g'(x) = -2\sin 2x \). Placing these into the Quotient Rule gives us the derivative of the inner function of the composition, which is paramount in the complete differentiation process.

Trigonometric identities simplify the calculus related to trigonometric functions. In this exercise, we utilized one of the fundamental identities: the cosine angle difference identity:

• \( \cos(A-B) = \cos A \cos B + \sin A \sin B \)

Applying this identity allowed us to transform the expression \( \cos x \cdot \cos 2x + 2 \sin x \cdot \sin 2x \) into a simpler form, specifically \( \cos(x-2x) \) or \( \cos(-x) \), which further simplifies to \( \cos x \) given the even property of cosine. Such transformations are crucial as they often simplify expressions, making the differentiation process cleaner and more manageable. Recognizing and using these identities well leads to more effective problem-solving in calculus, especially when mixed with functions that have trigonometric characteristics.

There are multiple differentiation techniques, and choosing the right one depends on the form of the function you are working with. For composite functions, like \( y = (\frac{\sin x}{\cos 2x})^3 \), the Chain Rule is invaluable. Here's how it works:

• The Chain Rule states: \( \frac{d}{dx} [f(g(x))] = f'(g(x)) \cdot g'(x) \).
• In our case, \( u = \frac{\sin x}{\cos 2x} \) and \( y = u^3 \).

First, we differentiate \( y \) with respect to \( u \), obtaining \( 3u^2 \). Then, we differentiate \( u \) with respect to \( x \) using the Quotient Rule. Finally, we multiply these results according to the Chain Rule to find \( \frac{dy}{dx} \). Each stage of the process supports the final outcome in simplifying and correctly finding the derivative of complex composed functions. It emphasizes a deeper understanding of how differentiated rates of change combine in nested functions.