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The chain rule is an essential tool in calculus for finding the derivative of composite functions. Let's break it down simply: suppose you have a function within another function, like an onion with multiple layers.
Imagine a sandwich where the meat (inner function) is wrapped by the bread (outer function). To find the rate of change (derivative) of this entire sandwich (composite function), you need to determine the derivative of the outer layer first while keeping the inner layer unchanged.
Then, you find the derivative of the inner layer itself. Finally, you multiply these two derivatives together. This approach allows you to effectively "peel back the layers" of the sandwich (or onion) to understand how the entire function changes.

• The outer layer in this example is: \(u^{1/3}\)
• The inner layer is: \(u = 2r - r^2\)
• The result is: Multiply the derivatives of the outer and inner functions.

When differentiating composite functions, you apply the chain rule, as discussed. Composite functions are functions made up of more than one simpler function.
By differentiating each part of the function layer by layer, you start by focusing on the most outside layer, gradually working your way inward.
In practice, you apply the rule: identify the roles of each function - the inner and the outer.
Then, differentiate the outer function first while holding the inner function constant. Once you've done that, differentiate the inner function.
For the exercise at hand, this involves recognizing that \( (2r - r^2)^{1/3} \) contains the inner function \(u = 2r - r^2\) and the outer function \(u^{1/3}\). These two roles precisely illustrate how composite functions are differentiated step by step by peeling each layer away using the chain rule.

The power rule is one of the fundamental rules in calculus, used to differentiate functions of the form \(x^n\). According to the power rule, you bring down the exponent as a multiplier and subtract 1 from the original exponent.
So, the formula for the derivative of \(x^n\) is \(nx^{n-1}\).
This rule simplifies the process of finding derivatives and is particularly handy when dealing with polynomial functions.
In the given problem, after rewriting the function to \( (2r - r^2)^{1/3} \), the power rule helps us find the derivative of the outer function. Here, \(n = 1/3\), therefore the derivative of the outer part becomes \((1/3)(u^{-2/3})\).
It's a quick way to tackle powers and makes complex differentiation tasks more manageable, working especially well when combined with other rules like the chain rule.