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The product rule is a fundamental concept in calculus used for finding the derivative of a product of two functions. It states that if you have two differentiable functions, say \(u(x)\) and \(v(x)\), then the derivative of their product is not simply the product of their derivatives. Instead, it follows a particular formula: \((u \cdot v)' = u' \cdot v + u \cdot v'\).
This means that to differentiate the product \(u(x) \cdot v(x)\), you need to compute the derivative of \(u(x)\), denoted as \(u'(x)\), multiply it by \(v(x)\), then add \(u(x)\) times the derivative of \(v(x)\), noted as \(v'(x)\).
This technique is crucial when dealing with products of functions that you encounter in calculus, especially when those functions are more complex like in the given exercise \(f(x) = \ln x \cdot \sin^{-1} x\). By applying the product rule, you are able to find the derivative \(f'(x)\) by carefully following the steps.

The natural logarithm, denoted as \(\ln x\), is an important mathematical function due to its natural occurrence in various fields like mathematics, engineering, and physics. The derivative of this function is one of its essential properties.
When differentiating \(\ln x\), the result is very straightforward: \(\frac{d}{dx}(\ln x) = \frac{1}{x}\).
This rule comes in handy when you encounter logarithmic expressions within more complex functions, such as in the terms of a product that also involves trigonometric or inverse trigonometric functions. Knowing this derivative rule allows you to tackle problems where \(\ln x\) is a component of a larger expression effortlessly.

Inverse trigonometric functions like \(\sin^{-1}x\) play a significant role in calculus as they are used to find angles with given trigonometric values. Differentiation of these functions allows us to understand how rapidly these angles change.
The derivative of \(\sin^{-1}x\), also known as the arcsine function, is \(\frac{d}{dx}(\sin^{-1} x) = \frac{1}{\sqrt{1 - x^2}}\).
This result is derived from the relationship between the arcsine function and the geometry of the unit circle, encapsulating how the rate of change of the angle slows as \(x\) approaches \(-1\) or \(1\). Using the derivative of the inverse sine is essential when dealing with compositions or products involving such functions, as seen in the problem of finding the derivative of \(f(x) = \ln x \cdot \sin^{-1} x\).