I've been studying how exactly Differential Calculus works by reading Mathematics: Its Content, methods and meaning.

The derivative provides a comparative indication of the rate of change at a particular point on a function on a non-linear function/graph. I find it helpful to think of the derivative as a property of change, i.e it quantifies a change so it can be compared with change occurring at another point of the function/graph.

The derivative is normally classified as instantaneous velocity when the graph is of distance with respect to time, i.e velocity is the derivative of distance with respect to time (\(v = s'(t) \)). The wording is sometimes confusing here. If velocity is the derivative of distance with respect to time, this is indicating that the distance is the measurement that is changing over time, i.e with respect to time. I suspect that the where this idea becomes generic is when the derivative can be defined using other independent variables (related of course) to measure the change of a magnitude with respect to another independent variable, which need not be time, as required for the velocity.

The derivative is the ratio of the difference of a function values at both ends of its domain, and the length of the that domain, and tends towards a value as the length of the domain shortens.

The derivative can be used to determine the differential of a function, which in turn can be used to estimate the increment of a function. I suspect that the differential and the increment are actually the same, however, there is a question of whether the differential calculation includes the infinitive term or not. If it does, then it's truly equal to the increment, if not, then it's only an approximation.

For example, \(\Delta y \approx f'(x)\Delta x\) while the increment is \(\Delta y\) as the infinitive term is not included, but when included accurately describes the increment of the function, ie \(\Delta y = f'(x) \Delta x + \alpha \Delta x\). Either way, the function differential helps determine functions that can be used to describe/define a natural phenomenon based on observed measurements represented as a graph (Differential equations). This in my mind is probably the most important application of differential calculus, because the applications are so diverse - you can mathematically model a phenomenon based on observations. 

The definition of the diffirential (of a funtion) is a finite magnitude for each increment of \(\Delta x\) and is at the same time proportional to \(\Delta x\). 

\(\Delta y = f'(x) \Delta x + \alpha x \)

So this essentially means that "a small increment of the function is equal to the differential of the function", meaning proportionate change in the function for the proportional change in x is the differential. The derivative is used to approximate the increment of a function: \(f'(x) \Delta x\)

The increment of the function or approximation of it (given by the diffirential) is useful at it provides the next value of the function given the change in the independant variable.. 

Orders of derivatives can be used in a polynomial to estimate/approximate/emulate a function when the derivatives of the function are known (Taylors formula/series). This is interesting because putting a function's derivatives into a polynomial can simulate that function, most accurately around the derivative, i.e where the change in x is the smallest. As the change in x grows larger, the Taylor series produces less accurate or more variance of the function produced, which then starts to deviate from the original function, ie the emulation becomes less correct. It appears from the theory that the higher the order of derivatives the more accurate this emulation. What this essentially allows us to do is approximate a function by using a polynomial expression.

The laws of nature, as a rule, can be expressed with good approximation by functions that may be diffirentiated as often as we like and that in turn may be approximated by polynomials, the degree of which determines the greater the accuracy.

The derivative is defined as the limit approached by the ratio of the change (distance/delta) of y to the change (distance/delta) of x, specifically as the change in x moves towards zero. Specifically, \(\lim\limits_{x \to 0} \frac{\Delta y}{\Delta x}\). For a while, I confused the notion that the derivative is somehow related to the ratio of the output of the function at a specific point and the input at that point. This is not correct as the derivative is specifically a ratio of the distance between two values of a function/curve (not the values themselves), to the distance or variance of the corresponding input values, which when taken to their limits, in no way relate directly to the value of the function at the derivative point, the derivative is itself unique indication of change, not function value.  

The derivative is derived from the result of the tan of the angle of inclination, where the change in y (distance) is the opposite side of a right-angled triangle and the change in x is the adjacent side of that triangle. The hypotenuse is the secant line that joins two points on the same function/graph and which becomes the tangent line as the change in x tends towards 0. This is how the derivative is actually defined and calculated, it is the definition of the tan of an angle, or more specifically and simply, the \(\frac{opposite}{adjacent}\) and this value/derivative changes as the change in x or the adjacent side becomes vanishingly small, which produces a vanishingly small \(\Delta x\) and correspondingly vanishingly small \(\Delta y\) or \(opposite\) side of the triangle. This reaffirms that the distances between the y-values and x-values are being considered and not the actual y-values and x-values themselves.

The mean-value theorem of Lagrange's formula allows one to calculate the exact increment of the function (not an approximation). More specifically and usefully it allows the exact determination of the actual value of any limit expression (which is impressive). Here a limit such as \(\lim\limits_{x \to 0} \frac{\phi '(x)}{\varphi '(x)} \) can be exactly defined to be \( \frac{\phi '(\varepsilon)}{\varphi '(\varepsilon)}\), meaning there exists a specific derivative at \(\varepsilon\) (on the same graph of as the function describing the limit) which is equal to the increment of the function exactly. Specifically between the ends of the range of the function. i.e \(\phi(b) - \varphi(a) = \phi '(\varepsilon)\) where the domain is (a, b), such that \( \frac{\phi(b) - \phi(0)}{\varphi(b) - \varphi(0)} = \frac{\phi `(\varepsilon)}{\varphi '(\varepsilon)}\)

The derivative is a limit, which hasn't got a defined value as the change in x is always tending towards zero and can do so indefinitively. Only when the change in x is factored out, can the derivative have a defined value. Every time you reduce the \(\Delta x\) within the difference equation, (\(\lim\limits_{x \to 0} \frac{f(x_0+ \Delta x) - f(x_0)}{\Delta x}\)), the derivative gets a different value, in most cases this becomes larger as the \(\Delta x\) decreases. As \(\lim\limits_{\Delta x \to 0}\) cannot ever be defined(there is always an ever-decreasing \(\Delta\)), the result of this expression or derivative is ever-changing.

The derivative can be depicted as \(f', f'(x)\), \(f^n(x)\), \(\lim\limits_{x \to 0} \frac{\Delta x}{\Delta y}\), \(\frac{\delta y}{\delta x}\) and \(\lim\limits_{x \to 0} \frac{f(x_0+ \Delta x) - f(x_0)}{\Delta x}\) where \(x_0\) is the initial point of the domain of the curve/function in question, and \(x_0 + \Delta x\) is the subsequent point on the domain of the same function. The distance between these two points is the \(\Delta y\)

The value of the derivative can indicate that the change that is occurring at a particular point in a non-linear function is either a) increasing, b) decreasing c) not changing. This can be used to determine the minima and maxima of that function. This is helpful and can be used to determine when, ie what input value i.e x will result in the largest, smallest value of the function. This is useful.

In calculating the derivative, the tangent can be defined because the derivative and the tangent are actually the same things, one is geometric and one is algebraic. To calculate the derivative requires establishing the relationship between the two secant points on the graph and manipulating the calculated relationship (\(\tan \alpha = \lim\limits_{x \to 0} \frac{opposite}{adjacent} \) or exactly the same is \(\tan \alpha \frac{\Delta y}{\Delta x}\)) by reducing the distance between them to such an extent (limit), that the geometric secant becomes a tangent and that limiting manipulation calculation becomes the derivative.

While the derivative is defined as \(\frac{\delta y}{\delta x}\) which is a limit more easily identifiable in this representation: \(\lim\limits_{\delta x \to 0} \frac{\Delta y}{\Delta x}\), these deltas are always decreasing as \(x \to 0\) but often the change in x is factored out of the difference equation (\(\lim\limits_{x \to 0} \frac{f(x_0+ \Delta x) - f(x_0)}{\Delta x}\)) by assuming x=0, this allows a definite value to be calculated. This is usually done by when using calulus 'first principles', i.e substitute the function you want to differentiate into the difference equation, ie (\(\lim\limits_{x \to 0} \frac{f(x_0+ \Delta x) - f(x_0)}{\Delta x}\))) to determine the derivative of any particular function. Rules have emerged when certain functions that are differentiated this way, leading to shortcuts which means that first principle calculation can be forgone by using the rules which result in the same result. These include the chain ruleproduct rule, and quotient rule etc.

The derivative is also interpreted as the slope of a non-linear function, in the same way that the slope is defined in a linear function, i.e for a linear function, the slope is defined as \(m = \frac{\Delta y}{\Delta x}\) and this is very similar to how the slope is defined in a non-linear function, however it is defined as a limit of this expression: \(\lim\limits_{\Delta x \to 0} \frac{\Delta y}{\Delta x}\).  

I've found some quotes to be especially helpful when studying calculus:

Wikipedia says: 

The slope of the tangent line is equal to the derivative of the funtion at the point the tangent line touches the graph.

The derivative measures the sensitiveity of change of the function value (y) with respect to the change in its argument (x)

Another informative quote from Mathematics: Its Content, methods and meaning is:

The derivative of a function \(y = f(x)\) of a variable x is a measure of the rate at which the value y of the funtion changes with respect to the chane of the variable x.

It is called the derivative of \(f\) with respect to \(x\). The derivative is the slope of the graph at each point in the graph.

For a function \(y = f(x)\), a small chane in \(x\), i.e \(\Delta x\) causes a change in y, i.e \(\Delta y\),

where: \(\Delta y = f(x_0 + \Delta x) - f(x_0)\) and \(\frac {\Delta y}{\Delta x} = f(x_0 + \Delta x) - f(x_0) x \frac{1}{\Delta x}\)

 Of differentiation, this is succinctly put:

The process of obtaining the derivative of a function by considering small changes in the function and in the independant variable, and finding the limiting value of the ratio of such changes.


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