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The Product Rule is a key tool in calculus for finding the derivative of the product of two functions. When you have two functions multiplied together, like in our problem with \(y = x^2 e^{-x}\), the Product Rule allows you to find the derivative efficiently. To apply the Product Rule, remember this simple formula:

• \((f \cdot g)' = f' \cdot g + f \cdot g'\)

This means you take the derivative of the first function \(f\), multiply it by the second function \(g\), and add the result to the first function times the derivative of the second. This ensures that you are accounting for all changes in the product as both functions vary. In our example, using the Product Rule requires identifying the two functions as \(f(x) = x^2\) and \(g(x) = e^{-x}\), then computing their derivatives separately to plug back into the rule.

The Power Rule is a simple yet very powerful tool used to differentiate functions of the form \(x^n\). The rule states:

• If \( f(x) = x^n \), then \( f'(x) = nx^{n-1} \).

This rule makes it straightforward to find the derivative of simple polynomial terms. In our original exercise, when differentiating \(f(x) = x^2\), you apply the Power Rule to get \(f'(x) = 2x\). Here, \(n = 2\), so the exponent is brought down as a coefficient, and the power of \(x\) is reduced by one, leading to an easy-to-remember pattern. Having such a rule at hand simplifies many differentiation problems by enabling you to instantly find derivatives of polynomial terms without much hassle. It's used frequently in conjunction with other rules, such as in our example with the Product Rule.

The Chain Rule is vital when differentiating composite functions - that is, functions nested within one another. It helps to "unravel" these layers efficiently. The Chain Rule can be understood as:

• If you have a function \( h(x) = f(g(x)) \), then the derivative \( h'(x) = f'(g(x)) \cdot g'(x) \).

In simple terms, differentiate the outer function at the inner function and multiply it by the derivative of the inner function. In the context of our example \(g(x) = e^{-x}\), this function can be expressed as a composition of \(e^u\) where \(u = -x\). Differentiating \(e^u\) gives \(e^u\) itself, and \(u' = -1\) (the derivative of \(-x\)). Thus, applying the Chain Rule gives \(g'(x) = -e^{-x}\). Recognizing when to use the Chain Rule is essential, especially in functions that include exponents, trigonometric, and logarithmic elements. Using the Chain Rule ensures you accurately capture all variabilities in multi-layered function expressions.