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When we talk about derivatives, we are essentially talking about the way a function behaves at a particular point. It's like zooming in infinitely close to a function's curve to see how it's changing exactly at a specific spot. The derivative of a function at a point is often compared to the slope of the tangent line to the curve at that point.

Mathematically, the derivative of a function \( f \) at a point \( a \), denoted \( f'(a) \), is defined by the limit:

• \( f'(a) = \lim_{h \to 0} \frac{f(a+h) - f(a)}{h} \)

This limit expression evaluates how much \( f(x) \) changes as \( x \) changes by a small amount \( h \).

Knowing the derivative definitions helps us understand the original problem because it sets a foundation for determining if the derivative exists for another related function, such as \((f+g)'(a)\). The existence of \( f'(a) \) and \( (f+g)'(a) \) implies that these functions have a defined rate of change at point \( a \).

The limit is crucial in calculus as it helps determine the behavior of a function as we approach a certain point. In simple terms, the limit describes what happens to the output of a function as the input gets very close to a particular value.

In calculus, the limit \( \lim_{h \to 0} \frac{f(a+h) - f(a)}{h} \) is used to define a derivative. This expression shows how the change in the function \( f \) as \( h \) approaches zero determines the slope of the function at \( x = a \). This concept plays a key role in understanding how various functions behave under infinitesimally small changes.

For the problem at hand, we separate the total limit of the function \((f+g)'(a)\) into parts, including \( \lim_{h \to 0} \frac{g(a+h) - g(a)}{h} \), which specifically addresses the function \( g \). If each part of the expression converges, then the overall limit does, suggesting that the derivative exists.

The sum of limits rule is a useful property in calculus that states the limit of a sum of functions is the sum of their limits. In equation form, this is expressed as:

• \( \lim_{h \to 0} [f(a+h) - f(a) + g(a+h) - g(a)]/h = \lim_{h \to 0} \frac{f(a+h) - f(a)}{h} + \lim_{h \to 0} \frac{g(a+h) - g(a)}{h} \)

This means each part can be examined individually, and if each part can be simplified, then the entire expression can be simplified as well.

In our particular exercise, we are given that both \( f'(a) \) and \( (f+g)'(a) \) exist. Thus by applying the sum of limits, we can separate the contributions of \( f \) and \( g \) to prove that \( g'(a) \) actually exists, confirming that \( \lim_{h \to 0} \frac{g(a+h) - g(a)}{h} \) is finite as well.

Proving the existence of a derivative like \( g'(a) \) through calculus often involves a structured approach using definitions and properties like limits. Let's understand how calculus proof helps:

• Start with what is known: Here, that \( f'(a) \) and \( (f+g)'(a) \) exist.
• Use definitions: This includes understanding derivatives as limits and breaking down expressions.
• Apply properties: Utilize sum of limits properties to dissect complex expressions.

For the problem in question, we use these steps:

We have \( (f+g)'(a) = \lim_{h \to 0} [f(a+h) - f(a) + g(a+h) - g(a)]/h \). If this sum of derivatives results in a defined number, and we subtract the known \( f'(a) \), the remaining term must also be a defined derivative, \( g'(a) \). Hence, the existence of \( (f+g)'(a) \) and \( f'(a) \) through calculus proof establishes the finite nature of \( g'(a) \). This is how calculus proofs solidify our understanding of such derivations.