91Ó°ÊÓ

In calculus, a function is said to be differentiable if it has a derivative at each point in its domain. This means the function's graph has a tangent at every point and does not have any sharp corners or breaks.
Differentiability is a stronger condition than continuity. While every differentiable function is continuous, the reverse isn't necessarily true. The function must be smooth and consistently changing for it to be differentiable.
For a function \( g(x) \) to be differentiable, \( g'(x) \), which is the derivative, needs to be defined everywhere in the domain of \( g(x) \). If we take the derivative \( g'(x) \) and integrate it as in the given exercise, it confirms that \( g(x) \) is differentiable because the derivative exists and can be processed further. This forms the foundation for solving calculus problems that involve integration and differentiation of functionals.

Integral calculus is one branch of calculus that is concerned with the concept of an integral. The integral can be understood as a representation of the accumulation of quantities. It is used to calculate areas under curves, among many other things.
In our exercise, we apply integral calculus to solve the problem \( \int \frac{g'(x)}{[g(x)]^2} \ dx \). This specifically involves using a pattern or a technique to simplify the integral. Recognition of the pattern allows us to use the formula \( \int \frac{f'(x)}{[f(x)]^2} \ dx = -\frac{1}{f(x)} + C \).
Here, the technique of recognizing a derivative within the integrand is a crucial skill. It simplifies problems and avoids lengthy computations of substitution or partial fraction scenarios. This demonstrates the power and elegance of integration techniques in integral calculus.

When performing indefinite integrals, the result is not a specific value but rather a family of functions. This is because integration is the reverse process of differentiation, and derivatives of functions differing only by a constant are the same.
The "constant of integration," denoted by \( C \), represents all possible constants that could be added to the function. It's an integral part (pun intended) of any indefinite integration. It reflects the fact that when we differentiate a constant, it turns to zero, so any constant could have been present originally.
In our solution for \( \int \frac{g'(x)}{[g(x)]^2} \ dx \), the addition of \( C \) in the final result, \( -\frac{1}{g(x)} + C \), ensures that any version of the antiderivative is represented. Don't forget that in real-world applications, this constant can often be determined by initial conditions or other specific restrictions in a problem.