91Ó°ÊÓ

Trigonometric identities play a crucial role in simplifying expressions involving trigonometric functions. In calculus, recognizing and applying these identities can simplify problems significantly. Here, understanding the identity \( \tan(x) + \cot(x) = \csc^2(x) \) is essential. This identity helps to convert the original expression \( y=\frac{\tan \frac{x}{2}+\cot \frac{x}{2}}{x} \) into a more manageable form. In this context, the cosecant squared function, \( \csc^2(x) \), represents the reciprocal of the sine squared function, \( \sin^2(x) \). Consequently, \( y \) transforms to \( y = \frac{\csc^2(\frac{x}{2})}{x} \) and further simplifies to \( y = \frac{1}{x \sin^2(\frac{x}{2})} \). This simplification is paramount for further steps like differentiation. By mastering these identities, you lay a strong foundation for solving complex trigonometric problems in calculus.

The chain rule is a fundamental tool in calculus for differentiating composite functions. It states that to differentiate a composite function, you must take the derivative of the outer function and multiply it by the derivative of the inner function. In our exercise, the function \( y = \frac{1}{x \sin^2(\frac{x}{2})} \) requires the use of the chain rule due to its nested structure.

• The outer function is \( y = \frac{1}{u} \) where \( u = x \sin^2(\frac{x}{2}) \).
• The inner function, \( u = x \sin^2(\frac{x}{2}) \), involves both basic multiplication and the squared sine function, which combines differentiation techniques.

The derivative of the outer function, involving \( \frac{1}{u} \), is \( -\frac{1}{u^2} \). For the inner function, you must utilize the product rule and recognize that \( \sin^2(\frac{x}{2}) = (\sin(\frac{x}{2}))^2 \). Differentiating these using standard techniques then applying the chain rule provides the complete derivative \( y' \). This comprehensive process underlines the power and necessity of the chain rule in tackling complex derivatives.

Differentiation is the process of finding the derivative, which represents the rate of change of a function with respect to a variable. In this exercise, we applied differentiation to the function \( y = \frac{1}{x \sin^2(\frac{x}{2})} \) to find \( y' \), its derivative. This process involved several steps including the application of trigonometric identities and the chain rule. The initial derivative, \( y' = -\frac{1}{x^2\sin^2(\frac{x}{2})} - \frac{x\cos(\frac{x}{2})}{\sin^3(\frac{x}{2})} \), can be seen as a combination of derivatives from basic trigonometric and algebraic principles.

• The first term in the derivative comes from differentiating the \( \frac{1}{u} \) form where \( u = x \sin^2(\frac{x}{2}) \).
• The second term appears from handling the trigonometric part \( \sin^2(\frac{x}{2}) \), which requires knowledge of the derivative of \( \sin \) and \( \cos \).

The simplification involving trigonometric functions like \( \csc \) and \( \cot \) often follows the differentiation process, making the final expression more compact and interpretable. Differentiation is not merely a mechanical process; it demands insight into intrinsic function relationships and transformations through calculus rules.