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When it comes to finding the derivative of a quotient (a fraction where the numerator and the denominator are both functions), the Quotient Rule is your go-to tool. Essentially, if you have a function that can be expressed as a division of two separate functions, say \( u(x) = \frac{g(x)}{h(x)} \), then the derivative of this function \( u'(x) \) can be found using the formula:

• \( u'(x) = \frac{g'(x)h(x) - g(x)h'(x)}{(h(x))^2} \)

The key to using the Quotient Rule effectively is to clearly identify which function is the numerator \( g(x) \) and which is the denominator \( h(x) \). After this, calculate their respective derivatives \( g'(x) \) and \( h'(x) \). Substitute these values back into the Quotient Rule formula to find the derivative of your original function. Remember to always simplify your final expression to make your answer more readable.

The Chain Rule is a fundamental principle in calculus when dealing with composite functions. If you need to differentiate a function that is nested within another function, the Chain Rule allows you to split the process into easier steps. Suppose you have \( y = f(g(x)) \), the Chain Rule states that the derivative \( y' \) is:

• \( y' = f'(g(x)) \cdot g'(x) \)

Essentially, you first take the derivative of the outer function \( f \) with respect to the inner function \( g(x) \), and then multiply it by the derivative of the inner function \( g(x) \) itself. In our exercise, when differentiating \( h(x) = \sqrt{x} \), you apply the Chain Rule by recognizing \( \sqrt{x} \) as \( x^{0.5} \), allowing a straightforward application of the power rule along with the chain effect. Using the Chain Rule ensures accuracy when dealing with layered functions, by systematically breaking down the differentiation process.

The Power Rule is a handy shortcut for taking derivatives of functions that are in the form of powers of \( x \). If you have a function like \( x^n \), the derivative is found by bringing down the power \( n \) as a coefficient and subtracting one from the power:

• \( \frac{d}{dx} x^n = nx^{n-1} \)

In the problem, \( g(x) = 4x^2 - x + 3 \) is an ideal candidate for the Power Rule. Differentiating each term:

• \( 4x^2 \) becomes \( 8x \),
• \( -x \) becomes \( -1 \),
• and the constant \( 3 \) becomes \( 0 \).

Thus, \( g'(x) = 8x - 1 \). The Power Rule is both simple and potent, allowing you to quickly find the derivative of polynomial terms without needing more complex rules.

After applying the Quotient Rule and other derivative rules, you often end up with complex expressions that need simplifying. Simplification involves several steps:

• Combining like terms,
• Reducing fractions,
• And cancelling common factors when possible.

In this exercise, after using the Quotient Rule, the expression obtained is:\[ f'(x) = \frac{8x^{1.5} -\sqrt{x} - 2x^{2} + 0.5x -1.5}{x} \]This expression can be simplified by dividing every term by \( x \) when appropriate, making it easier to understand and use. Simplification is an essential skill in calculus as it reduces the work for subsequent computations and helps in interpreting the result more intuitively. Always take care to simplify in steps to avoid errors, ensuring each part of the expression is accounted for.