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When confronted with a function that is formed by the division of two differentiable functions, finding the derivative may seem daunting. The derivative of a quotient is established using the quotient rule which enables us to differentiate such functions efficiently. The quotient rule states that for two differentiable functions, \( u(x) \) and \( v(x) \) where \( v(x) eq 0 \) and \( f(x) = \frac{u(x)}{v(x)} \) then the derivative is given by \( f'(x) = \frac{v(x) u'(x) - u(x) v'(x)}{(v(x))^2} \).
To effectively comprehend and apply this rule, one must ensure understanding of the numerator and denominator's individual derivatives, \(u'(x)\) and \(v'(x)\), respectively. Then, it's about combining these derivatives following the formula: multiply the derivative of the numerator by the denominator, subtract the product of the numerator and derivative of the denominator, and finally, divide by the square of the denominator. This process encapsulates the essence of the quotient rule.

The product rule is another vital calculus technique used when finding the derivative of two multiplied functions, \( u(x) \) and \( v(x) \), that are differentiable. The rule is concisely expressed as \( (u*v)' = u'v + uv' \), meaning that to find the derivative of the product, you take the derivative of the first function and multiply it by the second function as is, then add the product of the first function as is and the derivative of the second function.
In practice, the product rule is directly applicable when proving the quotient rule because it provides a foundation for handling the division of functions as multiplication with the reciprocal. The product rule for derivatives assures that each function partakes equally in the derivative, reflecting both their individual changes and their interaction within the product. This balanced consideration of both functions makes the product rule a cornerstone in understanding the behavior of complex functions under differentiation.

For a function to be considered differentiable, it must have a derivative at every point in its domain. Differentiable functions are smooth and do not have any sharp corners or discontinuities. This attribute is crucial because, in calculus, the derivative represents the rate of change of a function at a given point and is foundational to both the product and quotient rules.
Differentiability implies continuity; however, the converse is not always true—being continuous does not necessarily mean a function is differentiable. For example, a function could be continuous at a point but may have a cusp or vertical tangent at that point, preventing it from being differentiable there.
Furthermore, functions are required to be differentiable when applying calculus rules like the product and quotient rules. Therefore, when proving the quotient rule, one of the preliminary steps is to ensure that the two functions involved, \( u(x) \) and \( v(x) \) in this scenario, are indeed differentiable. This ensures that their derivatives exist and can be manipulated according to the rules, establishing the credibility of the results garnered from these calculus techniques.