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The derivative is a fundamental concept in calculus that measures how a function changes as its input changes. In simpler terms, it tells us the rate at which one quantity changes with respect to another. The derivative is crucial for understanding how a function behaves and is often represented by the symbol \( f'(x) \) for a given function \( f(x) \).

When calculating the derivative, we apply various rules depending on the structure of the function. The goal is to find the slope of the tangent line to the graph of the function at any given point. This helps us analyze and predict the function's behavior, such as identifying where it increases or decreases.

In the original exercise, we are tasked with finding the derivative of a function given the expression \( g(u) = \frac{2u^2}{(u^2 + u)^3} \). With more complex functions like this, we use certain rules like the quotient rule and the chain rule to break down the process into manageable steps.

The quotient rule is a formula used when taking the derivative of a quotient of two functions. It is particularly useful when you have a function that is structured as one part divided by another. The formula is as follows:

\[ h'(x) = \frac{f'(x)g(x) - f(x)g'(x)}{[g(x)]^2} \]

Here, \( h(x) = \frac{f(x)}{g(x)} \), where \( f(x) \) is the numerator function and \( g(x) \) is the denominator function. To apply the quotient rule, derive both the numerator and the denominator separately. Then plug those into the formula.

In the example provided, where \( f(u) = 2u^2 \) and \( g(u) = (u^2 + u)^3 \), we must first find \( f'(u) = 4u \). For \( g'(u) \), the chain rule is necessary since there is a composition of functions, leading us to another level of complexity, which we'll explore in the next section. Once both derivatives are found, they are substituted back into the quotient rule formula to get the solution.

The chain rule is applied when dealing with composite functions, where one function is nested inside another. It's an essential tool in calculus to differentiate these types of functions. The chain rule is expressed as:

\[ (f(g(x)))' = f'(g(x)) \cdot g'(x) \]

This formula indicates that to differentiate a composite function, you need to take the derivative of the outer function evaluated at the inner function and multiply it by the derivative of the inner function.

In this particular exercise, the function \( g(u) = (u^2 + u)^3 \) requires the use of the chain rule. The process involves recognizing that \( (u^2 + u) \) is the inner function while raising it to the power of 3 forms the outer function. We first differentiate the outer function \((u^2 + u)^3 \) as \( 3(u^2 + u)^2 \) and multiply it by the derivative of the inner function \( u^2 + u \), which is \( 2u + 1 \). These combined steps end in the solution for \( g'(u) \).

Function composition is when one function is applied inside another. This forms the essence of composite functions, where you take the output of one function and use it as the input for another. It is written as \((f \circ g)(x) = f(g(x))\).

In calculus, understanding function composition is crucial because it determines how to properly differentiate more complex, layered functions using rules like the chain rule.

• Inner Function: The function that is inside another function.
• Outer Function: The function that takes the inner function as its argument.

In our given problem, the function \( (u^2 + u)^3 \) is an example of a composed function. Recognizing this allows us to utilize the chain rule effectively, breaking down the problem into simpler parts which are more convenient to differentiate.
Understanding function composition helps in applying the correct differentiation technique, ensuring calculations are accurate and precise.