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Differentiable functions are a fundamental concept in calculus. A function is considered differentiable at a point, say at \( x = a \), if it has a derivative at that point. This means that the function's graph has a tangent line at \( x = a \), representing the function's instantaneous rate of change at that point. This characteristic ensures that the function is smooth and continuous without any sharp turns or discontinuities at \( a \).
Differentiability implies continuity, but the reverse is not necessarily true. For functions defined on an interval, they are differentiable in that interval if they are smooth across that whole interval without breaks or angles. This property is crucial when we work with polynomial approximations, as it allows us to use derivatives to understand the behavior of functions around a specific point.

Polynomial approximation involves representing a complex function with a polynomial, which is a simpler mathematical model. The Taylor polynomial, a common form of polynomial approximation, allows us to approximate differentiable functions near a specific point, \( x = a \).
The Taylor polynomial of degree \( n \) uses the function's values and its derivatives at \( a \) to create a polynomial \( g(x) \) that closely matches the original function \( f(x) \). This polynomial is constructed to have the same value and derivative as \( f(x) \) at \( x = a \), thus minimizing the approximation error near that point. It serves as a powerful tool in mathematical analysis and applications, such as in physics and engineering, where exact calculations with more complex functions might be impractical.

The limit of a function is a fundamental concept in calculus referring to the behavior of a function as the input approaches a certain value. In the context of polynomial approximation, limits help us assess how well our polynomial models the function it approximates.
Condition (ii) from the exercise asks us to consider the limit \( \lim_{x \to a} \frac{E(x)}{(x-a)^n} = 0 \). This condition ensures that the difference \( E(x) = f(x) - g(x) \) between the actual function and the polynomial becomes negligible compared to \((x-a)^n\) as \( x \) approaches \( a \). It suggests that the error in using the polynomial as an approximation becomes insignificant, ensuring \( g(x) \) offers a maximally accurate representation near the point \( x = a \).

Derivatives are critical in understanding the characteristics of functions, particularly their rates of change. When constructing Taylor polynomials, derivatives provide the needed information to achieve the best polynomial approximation.
The nth derivative of a function \( f(x) \) at a point \( a \) describes the behavior of the function around that point, and these derivatives are used to build the Taylor polynomial. In our exercise, the polynomial \( g(x) \) matches the function \( f(x) \) not only in value but also in all derivatives up to \( n \)-th order at \( x = a \). This means that the polynomials and functions share tangents and curvatures up to the nth degree, ensuring that the polynomial hugs the curve of the original function tightly around the pivot point \( a \).
In summary, derivatives equip us with the means to explore the nuances of a function's trajectory and use that information to make precise and accurate approximations.