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When you have a composite function, the Chain Rule is your go-to method for finding the derivative. Composite functions are just functions within other functions, like layers of an onion. Let's say you have a function denoted by \( f(g(x)) \). This means you apply the function \( g(x) \) first, and then apply the function \( f \) to the result. The Chain Rule helps you differentiate this complex function.

This rule says that the derivative of \( f(g(x)) \) is \( f'(g(x)) \cdot g'(x) \). You find the derivative of \( f \) at \( g(x) \) and multiply it by the derivative of \( g \) with respect to \( x \).

When working through differentiation, always look for opportunities to apply the Chain Rule as it simplifies finding derivatives considerably when dealing with nested functions.

The Absolute Value Function, denoted as \(|u|\), gives the size or magnitude of a number, regardless of its sign. This means \(|-3| = 3\) and \(|3| = 3\) as well. Absolute values are always non-negative.

For differentiability, it’s necessary to express the absolute value in a form that's easier to handle. This is where the expression \(|u| = \sqrt{u^2}\) becomes pivotal. It allows us to use calculus tools more effectively by converting the problem into a square root function.

Just remember: Absolute Value computations can sometimes lead to abrupt changes like sharp corners in a graph. Therefore, be careful when differentiating such functions, as they can affect the continuity and differentiability at specific points.

The square root function is crucial in calculus, especially when dealing with expressions like \(|u| = \sqrt{u^2}\). The square root function, represented as \( f(x) = \sqrt{x} \), is simply the positive number that, when multiplied by itself, gives \(x\).

When you're differentiating square roots, the derivative of \( f(x) = \sqrt{x} \) is \( \frac{1}{2\sqrt{x}} \). This formula is derived using the power rule for differentiation, where the square root can be expressed as an exponent: \( x^{1/2} \).

By understanding the behavior and derivative of the square root function, you can tackle a variety of calculus problems that involve transforming or simplifying expressions, like finding the derivative of an absolute value function.

In mathematics, derivatives represent how a function changes as its input changes. It is a fundamental tool in calculus and has numerous applications in various fields. Simply put, the derivative of a function gives us its rate of change or slope at any given point.

The process of finding a derivative is called differentiation. To differentiate complex functions, we combine various rules like the Chain Rule, Power Rule, and Product Rule. In our exercise, differentiation was used to find \( \frac{d}{dx} [|u|] \) using the expression \( |u| = \sqrt{u^2} \).

With practice, identifying which rule to apply and how to simplify expressions becomes easier. Always ensure that the function is differentiable at the point you are considering, and remember that derivatives provide a powerful way to understand the behavior of functions.