91Ó°ÊÓ

The derivative measures how a function changes as its input changes. In mathematical terms, if we have a function \( f(x) \), its derivative, denoted by \( f'(x) \) or \( \frac{df}{dx} \), tells us how \( f(x) \) changes with respect to \( x \). To find the derivative of a function, we usually follow a set of differentiation rules. With these rules, we can compute derivatives of simple polynomials and more complex functions involving products, quotients, or compositions of functions.

In our problem, we want to find the derivative of \( y = \left(x - \frac{1}{x} \right)^{-1} \). This involves applying differentiation rules carefully to each part of the function.

The chain rule is a fundamental technique in calculus used to find the derivative of composite functions. When we have a function \( y \) that depends on a variable \( u \), which in turn depends on another variable \( x \), the chain rule helps us link these dependencies. Mathematically, the chain rule is expressed as:

\[ \frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} \]

In the given problem, we have a nested function where \( y \) is a function of \( u \), and \( u \) is a function of \( x \). By setting \( u = x - \frac{1}{x} \), we simplify our differentiation process. We first find \( \frac{dy}{du} \) and then find \( \frac{du}{dx} \). Finally, we combine these derivatives to obtain \( \frac{dy}{dx} \) according to the chain rule.

An inner function is a term used to describe the function inside another function in composite functions. Identifying the inner function correctly is a crucial step when applying the chain rule. In our exercise, the inner function is given as:

\[ u = x - \frac{1}{x} \]

By recognizing this inner function, we simplify the original problem. We first differentiate \( u \) with respect to \( x \) to find \( \frac{du}{dx} \), giving us

\[ \frac{du}{dx} = 1 + \frac{1}{x^2} \]

Next, we differentiate the outer function (in this case, \( y \) in terms of \( u \)). Here, we have

\[ y = \frac{1}{u} \quad \text{and} \quad \frac{dy}{u} = -\frac{1}{u^2} \].

Now combining these steps with the chain rule gives us the final derivative.