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The Power Rule is an essential tool in calculus for taking derivatives, and it makes handling polynomial functions straightforward. The rule states that for a function of the form \(f(x) = x^n\), where \(n\) is any real number, the derivative \(f'(x)\) is given by \(nx^{n-1}\). This rule applies to each term individually.

To illustrate, consider the function \(u(x) = 6x - \frac{2}{x}\). We can rewrite \(\frac{2}{x}\) as \(2x^{-1}\) to apply the Power Rule easily. The derivative, then, becomes \(6 + 2(-1)x^{-2}\), which simplifies to \(6 - \frac{2}{x^2}\). This simple application of the Power Rule allows for tackling more complex tasks like finding derivatives in quotient functions.

The Quotient Rule is used when differentiating a function that is the ratio of two differentiable functions. This rule comes in handy to handle functions like \(f(x) = \frac{u(x)}{v(x)}\). Using the Quotient Rule, we find that the derivative \(f'(x)\) is \(\frac{v(x)u'(x) - u(x)v'(x)}{[v(x)]^2}\).

In the given exercise, the numerator \(u(x)\) is \(6x - \frac{2}{x}\) and the denominator \(v(x)\) is \(x^2 + \sqrt{x}\). Applying the Quotient Rule requires calculating both \(u'(x)\) and \(v'(x)\), which we have found using the Power and Chain Rules. Substituting these derivatives into the Quotient Rule provides the derivative of the entire function, allowing for comprehensive treatment in solutions of complex rational functions.

The Chain Rule is crucial when dealing with composite functions. It states that if a function \(y = f(g(x))\) where \(f\) and \(g\) are functions, the derivative \(y'\) is \(f'(g(x))g'(x)\). This rule helps dissect complex functions into simpler parts.

For instance, consider \(v(x) = x^2 + \sqrt{x}\). The term \(\sqrt{x}\) can be rewritten as \(x^{1/2}\). Using the Chain Rule, \(v'(x) = 2x + \frac{1}{2}x^{-1/2}\) gives us the derivative of the denominator function. This derivative is essential when used with the Quotient Rule to determine the overall derivative of combined and nested functions.

Simplification of expressions is a critical step in deriving final solutions, especially after using several differentiation rules. Once a derivative is calculated, simplifying it can aid in interpretation and further calculations.

After applying the Quotient Rule for the original exercise, you're faced with a complex expression. Simplification involves combining like terms and ensuring that the expression is presented in its simplest form. This might involve factoring expressions, canceling terms correctly, or rewriting terms for clarity.

• Ensure all terms are correctly distributed and combine any like terms.
• Seek common factors or denominators that can simplify overall presentation.
• Double-check all manipulations to avoid errors in final interpretations.

Through careful and methodical simplification, expression handling becomes more manageable and less prone to oversight or mistakes.