91Ó°ÊÓ

The chain rule is a fundamental concept in calculus used to differentiate composite functions. A composite function is when one function is applied inside another, like in our example where we have \( \sin(4x) \), which is a composition of \( \sin(u) \) and \( 4x \). The chain rule states that to differentiate a composite function, you first differentiate the outer function, leaving the inner function unchanged, and then multiply it by the derivative of the inner function.
For the function \( \sin(4x) \), you:

• Differentiated the outer function \( \sin(u) \), resulting in \( \cos(u) \).
• Next, multiply it by the derivative of the inner function, which is \( 4x \). The derivative of \( 4x \) is \( 4 \).

Hence, the derivative of \( \sin(4x) \) is \( 4 \cos(4x) \). This method makes handling complex functions much simpler and manageable.

Trigonometric functions like sine, cosine, and tangent are essential in mathematics, especially calculus. In this problem, we are dealing with the sine function. The sine function has a well-known adjacent function: its derivative, which is the cosine function. This relationship is pivotal when differentiating functions that involve sine.
In calculus, when you differentiate \( \sin(x) \) with respect to \( x \), the result is \( \cos(x) \). This derivative rule is crucial and forms the backbone of trigonometric differentiation. It is highly useful in various applications, from physics to engineering.
Key points to remember during such differentiation are:

• The derivative of \( \sin(x) \) is \( \cos(x) \).
• The cosine function, which is the derivative, shifts the angle's value into another periodic function form, demonstrating cyclical behavior.

Understanding these properties helps to apply them correctly when differentiating trigonometric functions.

The composition of functions combines two or more functions to create a single expression. This is the core of what makes the problem unique, as you must recognize that the function \( \sin(4x) \) is composed by inserting \( 4x \) into the \( \sin(u) \) function, where \( u \) is \( 4x \).
When dealing with compositions:

• Identify the outer and inner functions. In our case, \( \sin(u) \) is the outer function, and \( 4x \) is the inner function.
• Apply the chain rule accordingly by differentiating the outer function while treating the inner part as constant, then multiply by the derivative of the inner function.

This allows for streamlining the process of differentiation and gaining insight into more complex structures that functions can take. This technique is very advantageous and frequently appears in advanced calculus problems, making mastery of compositions and differentiation rules like the chain rule very important.