91Ó°ÊÓ

The chain rule is a fundamental concept in calculus used to differentiate composite functions, which are functions built from two or more combined functions. For example, in our exercise, the function is composed of a logarithm and a trigonometric function. The chain rule can be expressed as follows:

• If you have a composite function, say \( y = f(g(x)) \), the derivative \( \frac{dy}{dx} \) can be found by multiplying two derivatives: \( f'(g(x)) \) and \( g'(x) \).

In the context of our problem, we broke it down into parts:

• The outer function is \( \, \log(\sin x) \, \).
• The inner function is \( \, \sin x \, \).

To apply the chain rule, we first differentiate the outer function with respect to the inner function, resulting in a factor of \( \frac{1}{\sin x} \). We then differentiate the inner function, \( \sin x \), with respect to \( x \) to get \( \cos x \). Finally, these results are multiplied to find the complete derivative.

Trigonometric functions like \( \sin x \, \) (sine), \( \cos x \, \) (cosine), and \( \tan x \, \) (tangent) play an essential role in calculus, especially in this exercise.In the exercise, the function \( y = \log(\sin^2 x) \) involves taking the logarithm of a trigonometric expression. Here are a few important points about trigonometric functions:

• The sine function, \( \sin x \), oscillates between -1 and 1. It's important when incorporating periodic behavior in differentiation problems.
• The cosine function, \( \cos x \), is the derivative of \( \sin x \), and also oscillates between -1 and 1. Using it as \( \frac{d}{dx}(\sin x) = \cos x \) is crucial for chain rule applications.
• The cotangent function, \( \cot x = \frac{\cos x}{\sin x} \), often appears when working with derivatives of ratios involving sine and cosine.

Understanding how these functions interact and differentiate is key in solving problems like the one in our exercise.

Derivatives represent the rate of change of a function with respect to its variable. When we want to find the derivative of a function involving trigonometric and logarithmic expressions, using techniques like the chain rule and understanding how to handle each component is necessary.In the original problem, we needed to differentiate \( y = \log(\sin^2 x) \). We simplified it first to \( y = 2 \log(\sin x) \) using logarithmic properties.Key steps for finding derivatives:

• Understand the function type: Identify if the function is composite, involving multiple operations.
• Apply properties: Use log properties to simplify, if possible, before differentiating.
• Use differentiation techniques: Apply rules like the chain rule to handle composite parts.

Finally, compute the product of derivatives per the chain rule, leading in this instance to \( \frac{dy}{dx} = 2 \cdot \cot x \). Mastery of derivatives is essential for solving real-world and theoretical problems in calculus.