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Understanding derivatives is crucial in calculus and helps us find the slope of a function at any given point. The derivative essentially measures how a function changes as its input changes. In other words, it gives us the rate at which the function's value increases or decreases as the input varies. In this exercise, we start by finding the derivative of the function, which is necessary to determine the slope of the tangent line.To find the derivative of a function like \(f(x) = \frac{x^2 - 4}{5 - x}\), we rely on rules such as the power rule and, in this case, the quotient rule. Once we have the derivative, we can use it to calculate specific values, such as the slope at any point on the function's graph.

When dealing with the derivative of a quotient of two functions, the quotient rule is our go-to tool. The quotient rule states that for a function presented as \(\frac{u}{v}\), its derivative is \(\frac{u'v - v'u}{v^2}\). This rule is particularly handy when you have a more complex function, like a polynomial divided by another polynomial. For the given exercise, \(u = x^2 - 4\) and \(v = 5 - x\). We need to find the derivatives of both \(u\) and \(v\).

• First, calculate \(u'\), which is the derivative of \(x^2 - 4\). Using the power rule, we get \(u' = 2x\).
• Next, calculate \(v'\), the derivative of \(5 - x\), which is simply \(-1\).

Substituting these results into the quotient rule formula, we find the derivative of the original function, which is the first step towards finding the tangent line's equation.

The point-slope form is one of the key methods for writing the equation of a line, especially when dealing with lines tangent to functions at particular points. This form is expressed as \(y - y1 = m(x - x1)\), where \(m\) represents the slope of the line and \( (x1, y1) \) represents a point on the line. In tangent line problems, \(m\) is the derivative evaluated at a specific \(x\), giving us the slope of the tangent.For this exercise, once we've calculated \(f'(3)\), we insert it into the equation as the slope. Finding \(f(3)\) gives us the y-coordinate for our specific point of tangency, \( (x1, y1) = (3, f(3)) \). By substituting these values into the point-slope formula, it provides the precise linear equation for the tangent line at the point where \(x = 3\).

Function analysis involves studying the behavior of a mathematical function. This includes finding out how a function operates, determining where its maxima and minima are, and understanding how it changes. In the context of this exercise, analyzing \(f(x) = \frac{x^2 - 4}{5 - x}\) means understanding its rate of change and how it behaves at specific points. To analyze the given function:

• First, determine where it might be undefined, such as points where the denominator equals zero.
• Calculate the derivative to find how \(f(x)\) changes and at what rate.
• Use the derivative to find specific slopes along the function – crucial for finding tangent lines.

Function analysis provides deeper insights into the graph of a function beyond just finding derivatives and enables you to understand the full picture of the function's behavior.