91Ó°ÊÓ

Understanding the quotient rule of differentiation is essential when we want to take derivatives of functions that are represented as the ratio of two differentiable functions. In general terms, if you have a function that can be expressed as the quotient of two functions, say, \(u(x)\) and \(v(x)\), then the quotient rule is applied to find the derivative of the ratio \(\frac{u(x)}{v(x)}\).

The quotient rule formula states:
\[ (\frac{u}{v})' = \frac{v u' - u v'}{v^2} \]
where \(u'\) is the derivative of \(u(x)\) and \(v'\) is the derivative of \(v(x)\). It's a fundamental tool in calculus, as it simplifies the process of finding derivatives that may not be straightforward to differentiate using basic rules.

In our textbook exercise, we needed to differentiate \(f(x) = \frac{x^{2}}{x^{2}-4}\). Here, \(u(x) = x^2\) and \(v(x) = x^2 - 4\) are both differentiable functions. By applying the quotient rule, the derivative \(f'(x)\) is calculated as \(\frac{-8x}{(x^{2}-4)^{2}}\), shedding light on the rate of change of \(f\) with respect to \(x\).

A function is said to be continuous at a point if there are no interruptions, jumps, or breaks in its graph at that point. Conversely, a discontinuous point is where these interruptions occur. Identifying discontinuous points in a function is a critical step in the process of determining differentiability because a function must be continuous at a point in order to be differentiable there.

For the function \(f(x) = \frac{x^{2}}{x^{2}-4}\), we identify discontinuous points by setting the denominator equal to zero and solving for \(x\), which yields \(x = -2\) and \(x = 2\). These points are discontinuous because the function cannot be defined there—the division by zero is not allowed in mathematics. Therefore, we can conclude that \(f(x)\) is not differentiable at these points.

Remember, continuity is a prerequisite for differentiability, but the reverse is not necessarily true; there are continuous functions that are not differentiable. However, in this case, the discontinuities clearly signal a lack of differentiability at \(x = -2\) and \(x = 2\).

The derivative of a function represents the rate at which the function's value changes with respect to changes in the variable. It is a core concept in calculus, as it gives us insights into the behavior of functions, such as the rate of change, or slope at any given point on a curve.

In differentiation, when we calculate the derivative of a function \(f(x)\), denoted by \(f'(x)\), we are finding a new function that gives the slope of the tangent to the curve at any point \(x\). For the given function \(f(x) = \frac{x^{2}}{x^{2}-4}\), the derivative, as per the quotient rule, is \(f'(x) = \frac{-8x}{(x^{2}-4)^{2}}\).

This derivative tells us how \(f(x)\) behaves at each point except for where it's not defined or continuous. It is differentiable everywhere except at the discontinuous points \(x = -2\) and \(x = 2\), where the original function is undefined. Knowing the derivative allows us to analyze the function's increasing and decreasing behavior, as well as identify potential maxima, minima, and inflection points, which are valuable in both theoretical and applied mathematics.