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The quotient rule is an essential tool in calculus when you're dealing with derivatives of rational functions—that is, functions expressed as one function divided by another. It's like a sibling to the product rule, specifically tailored for division. This rule helps to efficiently calculate the derivative of a quotient of two differentiable functions.

To apply the quotient rule, if you have a function of the form \( \frac{f(x)}{g(x)} \), then its derivative is calculated as:

• First, find the derivative of the numerator, \( f'(x) \).
• Then, find the derivative of the denominator, \( g'(x) \).
• Use the formula: \[ \frac{f'(x)g(x) - g'(x)f(x)}{[g(x)]^2} \]

This might seem challenging at first, but with practice, it becomes straightforward, just like in the example where we determined the derivative of \( \frac{3x-2}{5x+1} \) using the quotient rule. Applying the rule, we plugged in the derivatives \( f'(x) = 3 \) and \( g'(x) = 5 \), and smoothly came to the next part of the solution.

Differentiation is one of the core concepts in calculus and it involves finding the rate at which a function is changing at any given point. This rate of change is represented by the derivative. Under the hood, differentiation tells us how a small change in the input of a function impacts the function's output.

In simple terms, for a function \( f(x) \), differentiation aims to find its derivative, noted as \( f'(x) \). This can be easily achieved using basic differentiation rules, like the power rule, which we used in our example.

• The power rule states that for a power \( x^n \), the derivative is \( nx^{n-1} \).

Applying this to \( f(x) = 3x - 2 \), we find \( f'(x) = 3 \) because \( x^1 \) becomes \( 1 \cdot x^{0} = 1 \). A similar approach finds \( g'(x) = 5 \) for \( g(x) = 5x + 1 \). With these derivatives, we are equipped to proceed using the quotient rule.

Simplifying a rational function often serves as the final step in calculus exercises like finding derivatives. In simplification, you aim to reduce the function into its simplest terms to either make it more interpretable or become easier to evaluate.

For example, after applying the quotient rule in the previous steps, we reached the expression:\[ \frac{15x + 3 - (15x - 10)}{(5x + 1)^2}. \]To simplify, notice that the terms \( 15x \) cancel out, leaving us with:

• Combining constants gives \( 3 + 10 = 13 \).
• Thus, the simplified derivative of the function is \( \frac{13}{(5x+1)^2} \).

This resulting expression conveys clearer information and can be easily interpreted or further used in other mathematical contexts like integration or limits.