91Ó°ÊÓ

The Quotient Rule is a fundamental technique in calculus used to find the derivative of the quotient of two functions. In the given exercise, we have the function \(f(x)=\frac{(x+1)(x-2)}{x^{2}-5x+1}\). This function is composed of a numerator function \((x+1)(x-2)\) and a denominator function \(x^{2}-5x+1\). The Quotient Rule states that the derivative of a quotient \(\frac{u(x)}{v(x)}\) is given by the formula:

• \[ f'(x) = \frac{u'(x)v(x) - u(x)v'(x)}{[v(x)]^2} \]

Here, \(u(x)\) is \((x+1)(x-2)\), and \(v(x)\) is \(x^{2}-5x+1\). The first step involves identifying \(u'(x)\) and \(v'(x)\), which are the derivatives of the numerator and denominator respectively. After calculating these, as shown in the solution, we substitute them into the formula to find the derivative of the entire function. This requires careful application of calculus principles to ensure accuracy and understanding. The result is the derivative of the original function, which provides information about its rate of change.

The Product Rule is essential when differentiating expressions that are the product of two functions. In this exercise, we first need to compute the derivative of the numerator \((x+1)(x-2)\). This requires the use of the Product Rule, which states:

• \[ (uv)' = u'v + uv' \]

Here, \(u(x) = x+1\) and \(v(x) = x-2\). To find the derivative of the product, we first find \(u'(x) = 1\) and \(v'(x) = 1\), then substitute them back using the Product Rule formula:

• \[ (x+1)'(x-2) + (x+1)(x-2)' = (1)(x-2) + (x+1)(1) \]

Simplifying this, we find the derivative of the numerator to be \(2x-1\). This calculated derivative is an essential component when we employ the Quotient Rule for our entire function.

Finding derivatives is a crucial aspect of calculus, involved in understanding the instantaneous rate of change of functions. Derivatives reveal the behavior and slope of a function at any given point. When dealing with complex functions, such as quotients or products, calculus rules like the Quotient and Product Rules come into play.To find derivatives, you typically:

• Identify which rule applies (e.g., Power Rule, Product Rule, Quotient Rule).
• Compute the necessary individual derivatives (e.g., of the numerator and denominator separately in quotient scenarios).
• Substitute these derivatives into the appropriate formula to get the derivative of the entire function.

This systematic approach is not just about computation, but also about understanding how functions behave. It enhances problem-solving skills and the ability to analyze mathematically complex scenarios. In the solution provided, these steps collectively allow you to find the derivative of the function \((x+1)(x-2)/(x^{2}-5x+1)\), providing insight into its rate of change.