91Ó°ÊÓ

The concept of a derivative is fundamental in calculus as it represents the rate of change of a function with respect to its variable. Essentially, the derivative tells us how a function changes at any given point. For the function \( f(x) \), its derivative \( f'(x) \) is a new function that gives the slope of the tangent line to the graph of \( f(x) \) at any point \( x \).

Here are some key points to remember about derivatives:

• They are used to find the instantaneous rate of change, such as velocity or acceleration.
• In a graphical sense, they represent the slope of the tangent line to the function at a particular point.
• A critical application of derivatives is in finding the extreme values (maximums and minimums) of functions.

In the given exercise, the derivative \( f'(x) = \frac{2x \cos(2x) - \sin(2x)}{x^2} \) was found. We used this to locate the extreme values of the function, which are points where the function reaches its highest or lowest values within a given range.

The Quotient Rule is a technique used in calculus to find the derivative of a function that is the division of two other functions. When we have a function in the form \( \frac{u(x)}{v(x)} \), where both \( u(x) \) and \( v(x) \) are differentiable functions, the derivative can be found using the Quotient Rule.

The Quotient Rule states:\[\left( \frac{u}{v} \right)' = \frac{v \cdot u' - u \cdot v'}{v^2}\]where \( u' \) and \( v' \) are the derivatives of \( u \) and \( v \), respectively.

In the exercise, we applied the Quotient Rule to the component \( \frac{\sin(2x)}{x} \) as part of the process to find the derivative of \( f(x) = 3 - \frac{\sin(2x)}{x} \). This rule is essential when dealing with functions divided by one another, ensuring we correctly differentiate them so that we can analyze their behavior thoroughly.

The Chain Rule is another pivotal concept in calculus that allows us to differentiate composite functions. A composite function is one where the output of one function is used as the input of another, symbolically expressed as \( h(x) = g(f(x)) \).

The Chain Rule states:\[h'(x) = g'(f(x)) \cdot f'(x)\]In simpler terms, the derivative of a composite function is the derivative of the outer function evaluated at the inner function times the derivative of the inner function.

In the context of our problem, the Chain Rule was applied when taking the derivative of \( \sin(2x) \). The derivative of \( \sin(2x) \) is \( 2\cos(2x) \) because of the presence of the inner function \( 2x \), with its derivative being \( 2 \).

Understanding and applying the Chain Rule is crucial when dealing with nested functions, ensuring we accurately determine how these functions change.