91Ó°ÊÓ

In calculus, the derivative of a function is a fundamental concept that measures how a function’s output changes as its input changes. It is often represented as \( f'(x) \) or \( \frac{df}{dx} \) . The derivative helps us understand the rate of change at a particular point on a graph, which is the slope of the tangent line to the function at that point.

For example, to find the derivative of a simple power function like \( f(x) = x^n \), we use the power rule which states that \( f'(x) = nx^{n-1} \). However, if a function is a ratio of two different functions (a quotient), such as \( f(x) = \frac{1}{x-1} \), we must use the quotient rule to find its derivative, which leads us to our next critical topic.

The quotient rule is a technique in calculus used to find the derivative of functions that are ratios of two differentiable functions. The formula for the quotient rule is given by

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

where \( u \) and \( v \) are functions of \( x \), and \( u' \) and \( v' \) are their respective derivatives. When applying the quotient rule to the function \( f(x) = \frac{1}{x-1} \), we identify \( u = 1 \) and \( v = x - 1 \), and their derivatives \( u' = 0 \) since 1 is a constant, and \( v' = 1 \) since the derivative of \( x \) is 1.

Applying the quotient rule as shown in the textbook solution allows us to find the derivative that is needed to determine the slope of a tangent line at a specific point on the function's graph.

The point-slope form of a linear equation is a way to write the equation of a line when you know the slope and a point through which the line passes. It is written as:

\[ y - y_1 = m(x - x_1) \]

where \((x_1, y_1)\) is the point on the line, and \(m\) is the slope of the line. In our case, we have the point \((-1, -1/2)\) and the slope \( -1/4 \), calculated as the value of the derivative at \(x_1 = -1\).

This form is particularly useful because it’s straightforward to plug in the values and quickly derive the equation of the tangent line, as was demonstrated in the solution steps for finding the equation of the tangent line to the curve at the given point.

When sketching the graph of a function, we use the function’s properties, such as intercepts, asymptotes, and intervals of increase and decrease, to create an accurate representation of the function's behavior. Recognizing these features and understanding the behavior of a function near asymptotes are pivotal in sketching.

For the function \( f(x) = \frac{1}{x-1} \), we know there is a vertical asymptote at \(x = 1\) since the function is undefined when \( x - 1 = 0 \). The graph tends towards infinity as it approaches the asymptote from the left side and negative infinity from the right side.

Additionally, the tangent line, which we found by using the function’s derivative and the point-slope form, is drawn at the specific point we are interested in. This line just touches the function's graph at one point and its slope tells us the rate of change of the function at that specific point. This visual combination on the same set of axes provides a rich understanding of how the function behaves and changes — step 3 in our solution showcases this process.