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I know that this question maybe sounds pretty generic, but it's a curiosity that I have and I didn't found any answer yet.

I recently started studying calculus using this material where is said that "The fundamental idea of calculus is to study change by studying "instantaneous" change". So, its said that when you're trying to find a instantaneous speed you should use the derivative of delta Distance by delta Time and not the "common" equation (without the derivate).

My questions are: (1) why derivaties represents the instantaneous change in the system, (2) why the equation not-derived don't represent correctly that instantaneous change (considering that that still receives the instantaneous parameters for time and space) and (3) why should I use calculus?

Thanks.

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(1) When a quantity varies "all the time", you need to observe it during "infinitesimal" durations, where they stay "constant". Calculus allows you to extend formulas that work with constant quantities to variable ones.

(2) If you accelerate regularly from $0 \text{ km}/\text h$ to $100\text{ km}/\text h$ in one minute, what distance do you travel ? To answer that you need to take into account that the speed changes all the time, otherwise, you don't know what speed to use. By calculus, you prove that the average speed will do.

(3) It depends what you do in life.

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Example: When you drive in your car, then your speedometer usually tells you how fast you are going. Speed/velocity is defined as the rate of change of the position. You might measure how far your have driven from a given point. This distance will change over time because you are moving. The rate at which this measurement of position/distance changes is exactly the speed.

Distance/position is then usually (in the US) measured in miles. Speed is measured in miles per hour (mph). That is, speed is what tells you how many miles you are moving per hour. With a constant speed of $60$ miles er hour, in one hour you would have travelled a distance of $60$ miles.

This is just one example of the use of rate of change in every day life. There are many others such as: the rate at which and investment grows, the rate at which the temperature changes, the rate at which the speed is changing (this is called acceleration), etc.

To study these rates of changes you need calculus and derivatives. Why to you need this? Well, because the derive is defined as the rate of change. So b definition studying rate of change is studying derivatives. So, while you might think of derivatives as representing rates of change, they are really defined as such. If you don't want to use calculus and derivatives to talk about rate of change, then what do you want to do?

If you have a function that tells you the position of something (as a function of time), then the derivative will tell you the speed of that something (as a function of time).

Often we are more interested in how fast a quantity is changing that how the quantity is changing.

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(1) Consider the other side of the question for a moment: What is an average rate of change? This is when you consider the slope of a secant line that passes through your function at two points. If we (loosely) consider time as our independent variable, it is not possible to be in two points at once. Speed (as mentioned in other answers) is an excellent example of when instantaneous rates of change become useful: when driving a car, we often only care about our immediate velocity (interpreted as speed) which is the derivative of our position function. The attached graphic is a rudimentary example of position vs. derivative, or speed:

Position vs. Derivative

(2) To obtain a rate of change, something indeed needs to be changing. This "thing," let's again assume a position function, could have several changes in direction away and toward the axis (assuming univariate for simplicity). Now, note that averages are not reliable enough to gauge "how fast" the function is approaching or receding from the axis. Thus, to discover the immediate rate of change at a given time, we let our chosen point (in time, presumably) remain stationary while moving another arbitrary point along the function toward it. The line these two form along the function becomes a tangent when the two finally meet. This is described in the difference quotient:

$\lim\limits_{h\to 0}\frac{f(x+h)-f(x)}{h}$

Where $h$ is some arbitrary difference between your point of interest and the approaching point. As $h$ approaches zero, the secant connecting to two becomes a tangent.

(3) The "why" of calculus often boils down to two questions the deeper you go: do you know enough to support your desired occupation? Do you know enough to satisfy your personal interests. The first question is generally simpler to answer than the second, which generally leads to more questions.

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