# History of differentiation

#### TheSpArK505

Joined Sep 25, 2013
111
Hello guys.
I am googling about the history of differentiation, and in essence it says that differentiation was developed to find the tangent to a curve.
I know the definition of differentiation is about that but why, what problem that requires the potential of knowing the tangent of a curve in order to solve it ??

also, any link or article that gives a clearer idea would be appreciated.

• circuit lover

#### MrAl

Joined Jun 17, 2014
6,513
Hi,

Differentiation is so widely used it's in most physics problems.

One simple example is what direction does a ball on the end of a string fly when you spin it around in a circle fast and then suddenly let go. It flies out along a tangent line to the circle.

Another simple example is a stiff board that is placed so that it leans on a wheel and the wheel is on the ground and the other end of the board is on the ground. The wheel is the circle, and the board lies on a line that is tangent to the circle at the point where the board touches the wheel.

An even simpler example is where the board is laying on the ground and the wheel is placed on top of the board so it's circular edge is touching the board. The board is again tangent to the wheel at the point where the wheel touches the board. In this case the tangent line is at zero degrees because the board is perfectly horizontal: ____O____

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• circuit lover

#### wayneh

Joined Sep 9, 2010
16,102
... what problem that requires the potential of knowing the tangent of a curve in order to solve it ??
I can't answer this regarding the genuine history of differentiation, but I can tell you that mathematicians don't necessarily need the motivation of a real-world application in order to seek a solution to some problem. If they did, we'd call them engineers or even physicists. A mathematician does it because its there.

• cmartinez

#### amilton542

Joined Nov 13, 2010
496
An object of mass in free fall is accelerating at every point along the way. The tangent line to this curve, at a point of interest, gives the instantaneous velocity of the object at that moment. If this line is not tangent to this function f at the given point then it is a secant line and it will cut the curve somewhere else and will yield the average over some interval of time.

#### TheSpArK505

Joined Sep 25, 2013
111
So for example in general, when do I use differentiation or integration ?? like in the case of capacitor's current it is the derivative of voltage, and for inductor's voltage that equals to the integration of its current? is it related to a quantity that decreases with time-----differentiation.
And if it increases with time its integration !!

#### ErnieM

Joined Apr 24, 2011
7,996
You use the method appropriate to the task. For a capacitor, if you know the voltage you use differentiation to find the current. if you know the current you use integration to find the voltage. For many special cases there are shortcuts, such as when the voltage or current is sinusoidal.

An inductor is similar, just reverse voltage and current.

Don't look for some rule to blindly follow. Look at the math and understand what it tells you.

#### wayneh

Joined Sep 9, 2010
16,102
So for example in general, when do I use differentiation or integration ??
One is the 'opposite' of the other, so your question is a bit like asking the general rule for using a staircase: You go up when you want to go up, and down when you want to go down.

• atferrari and ErnieM

#### GopherT

Joined Nov 23, 2012
7,983
So for example in general, when do I use differentiation or integration ?? like in the case of capacitor's current it is the derivative of voltage, and for inductor's voltage that equals to the integration of its current? is it related to a quantity that decreases with time-----differentiation.
And if it increases with time its integration !!
When you look at distance, velocity, acceleration - you can see the relationship.
D = vt
v = at^2

Phase shifted sine/cosine relationships are not so easy to "see" but a trial and sanity check brings you a great reward of understanding.

#### cmartinez

Joined Jan 17, 2007
6,469
differentiation was developed to find the tangent to a curve
That's one of the many interpretations that differentiation has. But essentially, what it is is the expression of the instantaneous rate of change of one variable with respect to another. When you say that its purpose is finding the tangent to a curve you're limiting its definition to graphic applications.

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• wayneh

#### wayneh

Joined Sep 9, 2010
16,102
That's one of the many interpretations that differentiation has. But essentially, what it is is the expression of the instantaneous rate of change of one variable with respect to another. When you say that its purpose is finding the tangent to a curve you're limiting its definition to graphical applications.
I was assuming the TS may know more about the history than I do. I don't know what motivated the inventor(s). Are we talking about Newton and/or Leibniz? Those guys didn't care about tangent lines, per se.

#### cmartinez

Joined Jan 17, 2007
6,469
Newton and/or Leibniz
As a matter of fact, both. The two of them arrived at the same conclusions through different paths. One was limit theory, and the other infinitesimal differences. Curiously enough, it took both disciplines and some ingenuity to solve Zeno's paradox almost 2,500 years after it was first postulated.

#### crutschow

Joined Mar 14, 2008
23,378
in general, when do I use differentiation or integration ?
In general, you differentiate when you want to know how fast a parameter is changing at some instant in time.
You integrate when you want to sum a parameter over some length of time.
Thus the differential of current versus time (di/dt) tells us how much the current is changing with time.
And the integral of current over a time interval (∫i dt) tells how much charge has been transferred in that interval.
That's greatly simplified, of course.
is it related to a quantity that decreases with time-----differentiation.
And if it increases with time its integration !!
It has nothing to do with whether the quantity is increasing or decreasing.

• cmartinez

#### MrAl

Joined Jun 17, 2014
6,513
So for example in general, when do I use differentiation or integration ?? like in the case of capacitor's current it is the derivative of voltage, and for inductor's voltage that equals to the integration of its current? is it related to a quantity that decreases with time-----differentiation.
And if it increases with time its integration !!
Hi,

Since we were talking about differentiation i'll go with that first.

You use this when you want to know or describe the slope of the function at a given point. In electrical networks this is of prime importance because we find components that can be accurately described by a slope of some kind.
For the capacitor we have:
i=C*dv/dt

where the current 'i' is equal to the slope of the voltage 'v' with time 't' times the value of the capacitor 'C'.
To be succinct, this gives us the current at a given point in time knowing the rate of change of voltage at that time and the value of the capacitor. This is so important in electrical circuits because sometimes we just dont know anything else but we still want to solve the circuit problem. Knowing this relationship between 'i' and 't' allows us to solve problems that we might not be able to solve otherwise.

In the simplest case, if 'v' is changing as a straight line then we have something like:
v=m*t+b

where m is the slope, which is the same as the derivative. When we take the derivative with respect to time, we get:
dv/dt=m

which is the slope as expected.

Now that we know what dv/dt is, we can use:
i=C*dv/dt

directly and get:
i=C*m

and since 'C' is a constant and 'm' is a constant (in this example only) we know that 'i' is a constant also. So we know now that in this example 'i' was a constant current of value C*m.

That's the simplest example i think. Other examples are a little more complicated so 'i' may not work out so simple, but it's the same idea.

It also helps to look at a numerical derivative which for a smooth curve could be found from:
dv/dt=(v2-v1)/(i2-i1)

where v1 and v2 are the voltages measured at times t1 and t2 respectively. This gives an approximation to the exact derivative we found far above. There are many algorithms that are based on this simple example and are used in almost every circuit simulator program. They get more complicated though because they are seeking to get more accurate results. Taylor's is one such method which is theoretically extendable to high order.

#### WBahn

Joined Mar 31, 2012
24,709
So for example in general, when do I use differentiation or integration ?? like in the case of capacitor's current it is the derivative of voltage, and for inductor's voltage that equals to the integration of its current? is it related to a quantity that decreases with time-----differentiation.
And if it increases with time its integration !!
Integration is used to determine the overall net effect of something over some interval (be it time, distance, or whatever). Differentiation is used to determine the rate at which one thing is changing with respect to another at a particular point (be it time, distance, or whatever).

My understanding is that Newton developed calculus as a geometrical tool for working with astronomical problems, particularly planetary orbits. Leibnitz developed it more as a pure mathematical construct for studying graph theory. Both developed it essentially independently at right about the same time with little knowledge of the other's work. Today we use mostly Leibnitz's notation because it was more mathematically consistent and elegant (probably due directly to his motivations).

http://bfy.tw/AFX6

This stuff is not hidden in the dark recesses of some dusty library basement. Please at least TRY to search out information so that at least you can start your query on a forum from a more focused perspective.

• cmartinez

#### amilton542

Joined Nov 13, 2010
496
An interesting premise in differential calculus is the extreme value theorem. It's one of my favourites because differentiation is masked by some rate at which something is changing but here it yields a maximum/minimum.

#### WBahn

Joined Mar 31, 2012
24,709
An interesting premise in differential calculus is the extreme value theorem. It's one of my favourites because differentiation is masked by some rate at which something is changing but here it yields a maximum/minimum.
Huh?

As I understand it, all the extreme value theorem says is that any continuous function defined over a closed interval has both a minimum and a maximum value that it reaches at least once on that interval.

Assuming that's the theorem you are talking about, how is that a premise in differential calculus?

I also don't know what you mean when you talk about differentiation being "masked" by some rate at which something is changing.

#### circuit lover

Joined Feb 25, 2017
2
Hello guys.
I am googling about the history of differentiation, and in essence it says that differentiation was developed to find the tangent to a curve.
I know the definition of differentiation is about that but why, what problem that requires the potential of knowing the tangent of a curve in order to solve it ??

also, any link or article that gives a clearer idea would be appreciated.
Hi In my point of view, differentiation is nothing but a part of whole, like " an atom in a molecule ". It's opposite is integration means " whole of that part ".