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In calculus, derivatives play a crucial role in understanding how functions change. The derivative of a function gives us the rate at which the function's value changes with respect to its input. It is a foundational concept in calculus that provides insights into the behavior and properties of functions.

There are several key points to understand about derivatives:

• They indicate the slope of a function at any given point.
• They help find maxima and minima of functions, which are the highest or lowest points.
• They are used to determine the instantaneous rate of change, similar to how speed measures the rate of change of distance with time.

For the given function, we are looking at the derivative to determine how the function behaves as the variable changes. This involves using specific rules of differentiation, such as the chain rule, especially when dealing with complex or composite functions.

The chain rule is a fundamental tool in calculus for finding the derivative of composite functions. A composite function is a function made up of other functions. This rule allows us to differentiate functions that are nested inside one another.

To apply the chain rule, follow these steps:

• Identify the inner and outer functions within the composite function. In our case, the outer function is the sine function, and the inner function is the polynomial inside the sine.
• Differentially the outer function with respect to the inner function, treating the entire inner function as a single variable.
• Differentiate the inner function with respect to its variable.
• Multiply these derivatives together to get the derivative of the original function.

This approach is efficient when tackling complex functions and is crucial for handling any scenario where one function is inside another. Using the chain rule simplifies the calculation process significantly, ensuring accuracy and ease when working with more complicated derivatives.

Trigonometric functions like sine, cosine, and tangent are essential in calculus, frequently appearing in many real-world problems and mathematical equations. These functions model periodic phenomena, such as waves and oscillations.

For derivatives involving trigonometric functions, it's crucial to understand their unique properties:

• The derivative of \( \sin(x) \) is \( \cos(x) \).
• The derivative of \( \cos(x) \) is \( -\sin(x) \).
• Scaling these functions, such as in our outer function \( 2\sin(u) \), affects the magnitude but not the form. The rules still apply.

In the given exercise, the sine function is scaled and combined with a polynomial expression inside, making it a perfect candidate for using the chain rule. Understanding the derivatives of these basic trigonometric functions and their behaviors in different contexts is crucial for anyone studying calculus, making it easier to tackle a variety of problems involving oscillatory or wave-like functions.