91Ó°ÊÓ

Derivatives are a fundamental concept in calculus, representing how a function changes at any given point. They tell us the rate at which one quantity changes with respect to another. In the case of inverse hyperbolic functions, derivatives allow us to understand how these functions behave as we change their inputs slightly.
The derivative of a function, often denoted as \(f'(x)\) or \(y'\), provides a mathematical way to express this change. To find a derivative, we frequently use rules like the power rule, product rule, and more specifically, the chain rule, which is very useful for functions within functions.
In our exercise with \(y = \cosh^{-1}(3x)\), we want to find the derivative of this function. We started by expressing it in a different form using natural logarithms to facilitate easier differentiation. This transformation allows us to apply standard differentiation techniques more efficiently, eventually leading us to find the change rate with respect to \(x\).

The chain rule is a powerful tool in calculus used to differentiate composite functions, which are functions whose composition involves two or more functions. When you have a function inside another, like \(y = \cosh^{-1}(3x)\), the chain rule helps find the derivative efficiently.
In essence, the chain rule states that to differentiate a composite function \(f(g(x))\), you take the derivative of the outer function \(f\) with respect to the inner function \(g(x)\), and then multiply it by the derivative of the inner function \(g(x)\) itself. In formula terms, this is expressed as:

• \(\frac{dy}{dx} = f'(g(x)) \cdot g'(x)\)

For the given exercise, we applied this rule to our transformed function \(y=\ln(3x + \sqrt{9x^2-1})\). Here, \(f(u)\) is the logarithmic function and \(g(x) = 3x + \sqrt{9x^2-1}\). This systematic application of the chain rule helped simplify the differentiation process, ultimately contributing to the solution.

The natural logarithm, denoted as \(\ln(x)\), is the logarithm to the base \(e\), where \(e\) is an irrational constant approximately equal to 2.71828. It plays a pivotal role in calculus and many scientific calculations due to its unique properties.
In differentiating functions involving natural logarithms, such as the transformed \(\cosh^{-1}(3x)\) to \(\ln(3x + \sqrt{9x^2-1})\), knowing the properties of logarithms is key. The natural logarithm's derivative formula is straightforward and follows from the definition of logarithms. Specifically, the derivative of \(\ln(u)\) with respect to \(u\) is \(\frac{1}{u}\).
This property facilitates the differentiation process when a function is expressed in logarithmic form, making complex expressions easier to handle. By expressing \(\cosh^{-1}(3x)\) as a natural logarithm, we reduce the complexity, allowing for simple derivative extraction using known formulas and rules like the chain rule to handle the composite structure.