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Understanding how to find the derivative of a function is a fundamental concept in calculus. Derivatives represent the rate of change of a function with respect to a variable, which is often time or space. Conceptually, finding a derivative can be thought of as finding the slope of the tangent line to the function curve at any point.To find the derivative of a function, follow these general steps:

• Identify the function you want to differentiate.
• Apply differentiation rules, such as the power rule, product rule, or quotient rule, depending on the form of your function.
• Simplify the result, if necessary.
• Evaluate the derivative at a specific point if required, as was the case when evaluating at \(x = 1\).

Finding derivatives is crucial in many real-world applications, such as finding velocities in physics or trends in economic data.In the exercise above, the derivative was found using the quotient rule because the function is given as a ratio of two polynomials. The derivative of the numerator and denominator was found separately and then combined using the rule.

Rational functions are expressions composed of ratios of polynomials. These can be written in the form \(\frac{P(x)}{Q(x)}\), where both \(P(x)\) and \(Q(x)\) are polynomials and \(Q(x)eq 0\). Rational functions appear frequently in calculus and have unique behaviors due to their denominator.Key characteristics of rational functions:

• Vertical Asymptotes: These occur at values of \(x\) that make the denominator \(Q(x)\) zero, unless cancelled by the numerator.
• Horizontal Asymptotes: These describe the end behavior of the function as \(x\) approaches infinity or negative infinity.
• Intercepts: Found where the function crosses the axes, with the x-intercept obtained by setting the numerator equal to zero, and the y-intercept found by evaluating the function at \(x = 0\) if possible.

In the given exercise, the rational function is \(\frac{4x+1}{x^2-5}\). Here, the differentiation process involves special rules because of the interaction between the numerator and denominator.

Calculus differentiation involves techniques to compute derivatives, an essential part of calculus that enables the understanding of changes within a system. It deals with various rules that simplify finding derivatives across different kinds of functions.Some important rules of differentiation include:

• Power Rule: For any function \(x^n\), the derivative is \(nx^{n-1}\).
• Product Rule: For two functions multiplied together, \(f(x)g(x)\), the derivative is \(f'(x)g(x) + f(x)g'(x)\).
• Quotient Rule: For a ratio of two functions \(\frac{u(x)}{v(x)}\), as seen in the exercise, the derivative is given by \( \frac{v(x)u'(x) - u(x)v'(x)}{v(x)^2} \).
• Chain Rule: Useful for finding the derivative of a composite function, where differentiating the outside function and then multiplying it by the derivative of the inside function is necessary.

The exercise at hand demonstrated the application of the quotient rule, providing a clear example of how to handle rational functions and compute their derivatives, thus illustrating a vital differentiation technique.