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The derivative of a function provides us with essential information about its behavior and how it changes at any given point. In mathematical terms, the derivative represents the rate at which a function is changing with respect to one of its variables, often denoted as \( x \). This makes derivatives incredibly useful when analyzing functions in calculus.

To find the derivative of a rational function, one often employs tools like the Quotient Rule. This rule is particularly handy when dealing with functions expressed as ratios of two other functions. The formula for the derivative of a quotient \( \frac{u(x)}{v(x)} \) is given by:

• \( u'(x) \): The derivative of the numerator.
• \( v'(x) \): The derivative of the denominator.
• Resulting in the formula, \( \frac{ u'(x) v(x) - u(x) v'(x) }{v(x)^2} \).

By applying this rule, one can accurately determine the derivative of complex rational functions.

Understanding whether a function is increasing or decreasing is crucial for comprehending its behavior over an interval. We can determine this behavior from the sign of the derivative:

When a derivative \( f'(x) \) is:

• Positive, the function is increasing on that interval.
• Negative, the function is decreasing on that interval.
• Zero, the function could either be at a maximum, a minimum, or staying constant.

In our original problem, the function \[ f(x) = \frac{3x + 5}{x + 1} \] was found to have a derivative \( f'(x) = \frac{-2}{(x + 1)^2} \).

Since the denominator \( (x + 1)^2 \) is always positive, and the numerator \(-2\) is always negative, \( f'(x) \) is negative for every value of \( x \). Therefore, we can confidently say the function is decreasing everywhere.

Simplifying derivatives is a fundamental skill in calculus that makes analyzing functions much easier. Once a derivative is found, simplifying it can help identify important characteristics without dealing with complex expressions.

In our example, after applying the Quotient Rule, we have:

• The expression \( f'(x) = \frac{3(x + 1) - (3x + 5)}{(x + 1)^2} \)
• Simplifying the numerator gives \( 3x + 3 - 3x - 5 = -2 \)
• So the simplified form is \( f'(x) = \frac{-2}{(x + 1)^2} \)

This simplification makes it easy to see that the function's derivative remains negative, indicating the constant decreasing nature of the function. Such simplifications help in gaining a clear insight without unnecessary complication.