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The chain rule is a powerful tool in calculus used to differentiate composite functions. It is especially helpful when dealing with nested functions, where one function is inside another. In this problem, the function involves both inverse hyperbolic functions and square roots. To tackle these, we must understand how the chain rule connects the outer and inner functions for effective differentiation.

The basic idea is to first differentiate the outer function while keeping the inner function intact, and then multiply by the derivative of the inner function. This ensures we account for all parts of the composite function.

For example, if you have a function like \\( f(g(x)) \), you would first find \\( f'(g(x)) \)and then multiply it by \\( g'(x) \). Applying this systematically helps break down complex expressions into manageable pieces.

Inverse hyperbolic functions are similar to inverse trigonometric functions but are derived from hyperbolic functions. These often appear in calculus problems due to their connections to algebraic and geometric concepts.

The inverse hyperbolic secant function, written as \\( \operatorname{sech}^{-1}(x) \), is a key component in this problem. Differentiating inverse hyperbolic functions may seem daunting, but by utilizing known formulas, it becomes straightforward.

• The derivative of \\( \operatorname{sech}^{-1}(u) \) is \(-\frac{1}{u \sqrt{1-u^2}} \).
• For our term \\( a \operatorname{sech}^{-1}\left(\frac{x}{a}\right) \), we differentiate based on the resulting formula from the chain rule, leading to the expression \\(-\frac{1}{x \sqrt{1-(x/a)^{2}}} \).

By leveraging these derivatives, we streamline the process of working with inverse hyperbolic functions in calculus.

Solving calculus problems requires systematically breaking down the expressions using known rules and methods. This problem, involving differentiating the equation of a tractrix, illustrates the need for precision and accuracy at each step.

First, recognize the structure of the function and identify all components and operations involved, such as inverse hyperbolic functions and square roots. Then, apply relevant rules like the chain rule to find individual derivatives.

• For compound expressions, differentiate each part separately. Here, separate the task into two main components: differentiating \\( a \operatorname{sech}^{-1}\left(\frac{x}{a}\right) \) and \\(-\sqrt{a^2-x^2} \).
• After finding the derivatives of each part, combine them to get the final derivative as shown: \\(-\frac{1}{x \sqrt{1 - (x/a)^2}} + \frac{x}{\sqrt{a^2-x^2}} \).

Approaching problems systematically with these strategies makes complex calculus problems more manageable and understandable.