I had some trouble in understanding the meaning of existence of a derivative of a function, either its first or second derivative or derivatives of higher order (more confusion).
Why the existence of a derivative of a function is sought and what exactly does it convey?
The Meaning of the First Derivative :
The first derivative of the function $~f(x)~$, which we write as $~f'(x)~$ or as $~\dfrac{df}{dx}~$, is the slope of the tangent line to the function at the point $~x~$. To put this in non-graphical terms, the first derivative tells us how whether a function is increasing or decreasing, and by how much it is increasing or decreasing. This information is reflected in the graph of a function by the slope of the tangent line to a point on the graph, which is sometimes describe as the slope of the function. Positive slope tells us that, as $~x~$ increases, $~f(x)~$ also increases. Negative slope tells us that, as $~x~$ increases, $~f(x)~$ decreases. Zero slope does not tell us anything in particular: the function may be increasing, decreasing, or at a local maximum or a local minimum at that point.
The Meaning of the Second Derivative :
The second derivative of a function is the derivative of the derivative of that function. We write it as $~f''(x)~$ or as $~\dfrac{d^2f}{dx^2}~$. While the first derivative can tell us if the function is increasing or decreasing, the second derivative tells us if the first derivative is increasing or decreasing. If the second derivative is positive, then the first derivative is increasing, so that the slope of the tangent line to the function is increasing as $~x~$ increases. We see this phenomenon graphically as the curve of the graph being concave up, that is, shaped like a parabola open upward. Likewise, if the second derivative is negative, then the first derivative is decreasing, so that the slope of the tangent line to the function is decreasing as $~x~$ increases. Graphically, we see this as the curve of the graph being concave down, that is, shaped like a parabola open downward. At the points where the second derivative is zero, we do not learn anything about the shape of the graph: it may be concave up or concave down, or it may be changing from concave up to concave down or changing from concave down to concave up.
Note: At any point $~x~$ if the graph of the function goes from concave up to concave down at that point, or if the graph of the function goes from concave down to concave up at that point, then the graph of a function $~f(x)~$ has an inflection point.
Clearly then, an inflection point can only happen where at points where the second derivative is $~0~$, because otherwise the point would the graph would be either completely concave up or completely concave down at that point. Just like in the case of local maxima and local minima and the first derivative, however, the presence of a point where the second derivative of a function is $~0~$ does not automatically tell us that the point is an inflection point.
Ref.: https://math.dartmouth.edu/opencalc2/cole/lecture8.pdf