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I am only starting learning calculus and it's difficult for me to understand the main concept behind calculus ideas particularly differentiation I have searched many resources but most of them are very similar explaining things with words like "speed", "rate of change", "tangent" and "function change in respect to input change"... I know the rules of computation but the purpose is not very clear for me

I would be very grateful if somebody could help me grasp this concept and explain why one would want to compute the "rate of change" of a function and what exactly problem do derivatives solve.

Pretend that I am very stupid (unfortunately I am :) ) and don't use any abstract concepts (even if they are intuitive to a human being ) as "speed" if possible Thanks

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    $\begingroup$ "Speed" is an abstract concept? $\endgroup$
    – Jack M
    Commented Nov 2, 2013 at 23:00
  • $\begingroup$ In some way- yes. I understand speed through mathematical manipulation only $\endgroup$
    – user2284340
    Commented Nov 2, 2013 at 23:05
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    $\begingroup$ Well then I think you need to take a step back and look at things with a fresh perspective. Speed is one of the most intuitive, unabstract notions around. Don't you agree that a car on the highway is faster than a snail? Measuring speed is the purpose of derivatives, so I would suggest you start by getting comfortable with the notion of speed first. Do you at least understand the concept of measuring something's speed in, say, miles per hour? $\endgroup$
    – Jack M
    Commented Nov 2, 2013 at 23:11
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    $\begingroup$ I agree with you on new perspective. I can tell that a car is faster than a snail just by the distance and time. $\endgroup$
    – user2284340
    Commented Nov 2, 2013 at 23:18
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    $\begingroup$ One might ask, do we ever look at our speedometer? How come? We want to describe how fast something is moving. Or how fast something is changing. $\endgroup$
    – littleO
    Commented Nov 3, 2013 at 1:00

9 Answers 9

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Suppose $f(x)=x^3$.

Then $f(2)=8$ and $f'(x)=3x^2$, so $f'(2)=3\cdot2^2=12$.

That means when $x=2$ and $f(x)=8$, then $f(x)$ is changing $12$ times as fast as $x$ is changing.

So suppose $x$ goes from $2$ to $2.0001$, the change being $\Delta x=0.001$. Then $f(x)$ should go from $8$ to about $8.0012$, the change being about $\Delta f(x)=0.0012$, i.e. $12$ times as much. Why not exactly $12$ times as much? Because as $x$ changes from $2$ to $2.0001$, the derivative, thus the rate of change, does not remain exactly $12$.

(In fact, $f(2.0001)=2.0001^2 = 8.0012\ 00060001$, so $12$ times as much is pretty close.)

When calculus is taught to liberal arts majors, this kind of thing should be considered far more important than chanting "n x to the n minus one", which is what is typically taught. A standard calculus course for math majors was created, then watered down to get a calculus course for English majors, then very large numbers of the latter were encouraged to take calculus and told it would look good on their resumes. The learn to answer questions like "Find the derivative of $f(x)=\sec^3(5x+2)$" without finding out that differential calculus is about instantaneous rates of change or why it is important in the development of science and engineering over the past few centuries. Mathematicians feel forced to go along with the system, which must be maintained because those students bring in tuition money. Mathematicians who sit on curriculum committees in large departments are not the ones who are assigned the task of teaching first-semester calculus, and don't know what goes on there. The ones who do know are often less experienced and are not the ones who will develop alternative sorts of courses, and must devote their energies to publishing research so that they can keep their jobs. If you try to include things like this in a calculus course at the expense of chanting "n x to the n minus one", the sort of student who's there only to get a grade says "Will this be on the department's common final exam in this course? No? Then why are you wasting our time on it? My father donates a lot of money to this university and he will complain to the Dean about you." ONLY mathematicians can change this situation, so they cannot forever plead that they were only following orders.

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    $\begingroup$ Why the down-vote? It's a bit of a rant, but there's also a useful explanation of how derivatives can be used. $\endgroup$
    – bubba
    Commented Nov 3, 2013 at 1:38
  • $\begingroup$ PS: I should add that understanding of this point should also be considered more important that technical details when calculus is taught to physics majors or math majors, etc. But if the latter have even a shred of competence, one need not explain that to them. $\endgroup$
    – Michael Hardy
    Commented Nov 3, 2013 at 4:28
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The "rate of change" quantity is describing a function in a particular way: if the "rate of change" is large, then the function is increasing quickly, and the relationship is direct in this way. One of the primary uses is to find out when the function might "turn", or rather, when it stops going "down" and starts going "up" or vise-versa. When the derivative is zero, then the function is not increasing or decreasing, and we say (usually) that such a point is a "minimum" or "maximum" of the function.

Such a point is useful in determining where the zeroes or factors of a polynomial might be. If the function is not polynomic, the zeroes are often useful information anyways.

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If you will forgive the self-referencing, I wrote a blog post about some bits of calculus from what I consider an approachable perspective.

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    $\begingroup$ Thanks. I will read that... $\endgroup$
    – user2284340
    Commented Nov 3, 2013 at 2:54
  • $\begingroup$ It might be a good idea to add the relevant parts of your blog post it in this answer. $\endgroup$
    – Amal Murali
    Commented Nov 3, 2013 at 10:43
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There's lots of good reasons for why you should care about a function's derivative! Firstly, it gives you a notion of slope for non-linear functions, so that you can compare the relative "steepness" of different functions. This slope is incredibly useful, particularly in physics, because this slope allows you to go from knowing the position of a body at any time to knowing it's instantaneous velocity, acceleration, jerk, snap, etc.

Here's another important use. If you look at the graph of a continuous function, what do you notice about the tangent line to the graph, at the function's local maxima and minima? The slope of the tangent is 0! Being able to take a derivative of a function allows you to find where the function takes on maximum and minimum values.

Derivatives also allow you to approximate many functions. If you look at the tangent line $y = f'(a)x + b$ to a function $f$ at a point $(a, f(a))$, for $c$ close to $a$, $f(c)$ is not very far off from $f'(a)c + b$, so you can use your tangent line to make a good guess as to what values your function takes on. For infinitely-differentiable functions, you can use second, third, fourth, etc., derivatives to get incredibly accurate approximations. On functions like $e^x$ and $\sin x$ you can even get the exact value of the function at any point just from its derivatives.

I'm assuming you haven't studied integration yet, but once you do, you'll see that the derivative becomes even more powerful when paired with the integral.

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As a real world example of a usage of rate of change....

I work for a machine vision company where it is extremely common for the computer to find the edge of a physical object. An edge is defined as a change in contrast (aka white on one side, black on the other). If the image slowly transitions from white through gray to black then the edge is poorly defined; in other words it has a very low rate of change. If there is a quick transition from white to black, then the edge is well defined; it has a high rate of change.

If the rate of change is plotted on a xy graph, then there will be a peak value where the calculated edge is located within the image.

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  • $\begingroup$ Very nice example... $\endgroup$
    – user2284340
    Commented Nov 4, 2013 at 18:39
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Remember back to Algebra, when you found slopes of lines. (The $m$ in $y = mx + b$)

Differentiation does the same thing, only it's not limited to straight lines. It can be used on any curve.

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  • $\begingroup$ yes,I remember that but it what is the purpose of finding the tangent then? $\endgroup$
    – user2284340
    Commented Nov 2, 2013 at 22:51
  • $\begingroup$ The purposes are countless, but some that come to mind are: finding minimums and maximums, (in physics) finding velocity(speed) of an object from its position, and approximations. $\endgroup$
    – shade4159
    Commented Nov 2, 2013 at 22:58
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If you look at a certain point on a graph, how steep is the graph at that point? Take the derivative.

What if you want to find whether the graph is getting steeper or less steep at a certain point? Take the derivative of the derivative (called the second derivative).

What if you want to know whether how fast the steepness is changing at a certain point? Take the third derivative.

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Derivatives are important in many ways, and one big one is this: they are the basis for differential equations.

A differential equation occurs when a function is related to one or more of its derivatives (which can be of various orders, not necessarily first). An equation is formed which involves a function, and derived versions of that function.

Functions that are related to their derivatives occur in nature. For instance, a mass oscillating on a spring has an acceleration which depends on its position: this is because the acceleration depends on force, and force is proportional to the displacement. But acceleration is the second derivative of the position.

We can solve differential equations. When we have the solution, we can plug in some initial conditions, and the equation will exactly describe what the system will do, such as give the position of the mass on a spring as a function of time. Even if we are not interested in acceleration, acceleration is involved in giving us that solution.

Solving differential equations gives us the power to predict the behavior of systems (to the extent that real systems behave like models), which is useful in various kinds of science and engineering.

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Derivatives are used in "gradient descent" — real world task from machine learning.

Let's say you want to find a local minimum on an Earth surface (x, y) and you have a function which tells you the current (x, y) point height. You can think of it as an above the sea height of where you stand right now on the surface.

By taking a derivative of this function on this point (x, y) you can find the most efficient way to descend, make steps one-by-one this way, taking a derivative on each step to correct your heading.

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