What are differentials and why is dy/dx sometimes treated as a fraction?

[kashan123999]

I just couldn't grasp the idea what are differentials,intuition behind them,applications of differentials? can anyone thoroughly tell me about it please? and why sometimes dy/dx is taken as a fraction?

 

[UltrafastPED]

Here is a lecture on differentials, with applications:
http://tutorial.math.lamar.edu/Classes/CalcI/Differentials.aspx

The dy/dx notation is due to Leibniz, and encodes a number of useful ideas - that is, it is a memory aid.

Also see: http://www.math.sunysb.edu/~tony/archive/calc/97calc1/handouts/dydx.html

 

[kashan123999]

i still couldn't grasp the intuition of differentials sir,please elaborate more

 

[Mark44]

i still couldn't grasp the intuition of differentials sir,please elaborate more

Did you view the topics in the links that UltrafastPED provided? If so, do you have specific questions about them?

 

[kashan123999]

Did you view the topics in the links that UltrafastPED provided? If so, do you have specific questions about them?

those are plane mathematical definitions of differentials written in almost every book,i couldn't grasp the idea from that,can anyone explain it intuitively please?

 

[kashan123999]

can anyone explain it kindly?

 

[Mark44]

Your question is too broad. Our purpose here is not to spoon-feed you everything there is to know about differentials. If you have some specific questions about what you've read, please ask them. Otherwise, it's your responsibility to do some research on your own, either on the internet or in some calculus textbooks.

 

[caldweab]

can anyone explain it kindly?

Differentials explain how a dynamic system is behaving. For example you can model population growth by a differential equation and the solution to this differential equation will allow you to predict how a population will grow or decay in a given period. Differentials can also express things like diffusion as is the case in the heat diffusion equation in heat transfer. That's the simplest way I can put it. It is not a fraction either. I mean dy/dx means that Y is changing with respect to x

 

[UltrafastPED]

As Leibniz visualize it, a differential is a small increment along the line tangent to the function; so when you say "dx" it is an increment along the x-axis.

When you say "dy" it is an increment along the line tangent to the function y(x) at the point (x,y(x)).

These increments are "infinitesimal values" in the original view of Leibniz which had the exact ration dy/dx = y'(x).

Thus dy = y'(x) dx.

 

[kashan123999]

Your question is too broad. Our purpose here is not to spoon-feed you everything there is to know about differentials. If you have some specific questions about what you've read, please ask them. Otherwise, it's your responsibility to do some research on your own, either on the internet or in some calculus textbooks.

I don't think at least answering about the intuition of differentials is "that" broad or "spoon-feeding"..,i have done my own research and couldn't find it...I have read the formula for differentials that is dy=f'(x)dx and know the idea of using dy/dx as fractions... but couldn't grasp the intuitive notion of that formula/differentials and also why dx = Δx and dy is different than Δy...they might be easier to understand for some,but not for me...and i am a 12th grader actually so can't comprehend them unfortunately as my textbook is not comprehensive about it

 

[SteamKing]

There are some concepts which cannot be grasped simply by reading a lecture or having somebody describe it to you. You have to make a mental 'leap' yourself before the understanding comes. Like you, in high school, I wanted to learn calculus because I was going to engineering college. I studied calculus books on my own before my senior year in HS, I took calculus that year, and I also took a short summer course after graduating HS. Somewhere in all that course work, I grasped limits, differentials and integrals.

My only solid advice about differentials is to look at the problem of finding a tangent to a curve. IMO, this gives the best visual and geometric illustration of how Delta-x and Delta-y become dx and dy. Start off drawing a secant to a curve and then imagine the two intersection points merging into one point as Delta-x shrinks.

 

[Superposed_Cat]

Are differentials the same as derivatives?

 

[UltrafastPED]

Are differentials the same as derivatives?

Almost ... the derivative, y'(x)=dy/dx, is the slope of the curve y(x) at the point x.
Geometrically it is the slope of the line that is tangent too y(x) at point x.

The differential, dy, is the rate of change of the height of the curve, y(x), with small displacements of the argument, x. These small changes of x are the differential dx.

This geometric analysis is what lead to the Leibniz notation for the derivative, dy/dx.

It is most commonly used for linear approximations.

 

[Superposed_Cat]

Thanks, why do we integrate on differential equations then?

 

[Mark44]

Thanks, why do we integrate on differential equations then?

Because integration is essentially the inverse operation of differentiation. A differential equation involves one or more derivatives of some unknown function.

 

[UltrafastPED]

Thanks, why do we integrate on differential equations then?

I'll give an example where the "differential form" provides some guidance in setting up direct integrations to solve simple differential equations:

Suppose you have Hooke's law, F_spring = -kz, where z is the displacement. Then if there we ignore gravity Newton's Second Law of Motion, F=ma=m dv/dt, can be used along with Hooke's law to form a differential equation:

1) dv/dt = -k/m z.

Putting this into differential form gives:

2) dv = -k/m z dt

We can integrate the LHS, but the RHS is in terms of z and t, so we are stuck. Now go back the equation (1) and use the chain rule to modify the LHS:

3) dv/dt = dv/dz dz/dt, but dz/dt = v, the velocity, so we have:

4) v dv/dz = -k/m z, which in differential form is:

5) v dv = -k/m z dz; and in this form we can integrate both sides; the limits of integration for velocity must correspond to the limits for position - that is, the boundary conditions must correspond.

Assuming an initial velocity v0 at displacement z0, and current velocity v, position z we get:

6) 1/2 v^2 - 1/2 v0^2= -1/2 (k/m) z^2 + 1/2 (k/m) z0^2 or rearranging:

7) 1/2 mv^2 + 1/2 k z^2 = 1/2 mv0^2 + 1/2 k z0^2.

Equation (7) says that the sum of the kinetic and potential energy of the spring at any time is equal to the sum of the original values.


Note that the differentials which appeared in our expressions lead directly to the integrals.

This exercise shows the use of differentials as they were originally used in the Leibniz notation. In this case the differentials were "exact differentials", we could integrate them immediately.

Unfortunately most differential equations are not solvable by the method of direct integration, but this technique did give us the nomenclature. Most differential equations are expressed in the form of derivatives, but both notations are used.

 

[WWGD]

To summarize what many have said ** , the differential is an approximation to the change of the function along the tangent line/ tangent linear object (plane, etc.) . In small intervals, the differential (of a differentiable function) is as accurate an approximation to the actual change Δf of the function as you want, in a precise ε-δ sense, i.e., if you want |Δf-df|<ε , then you can find a δ so that in an interval of radius δ where that condition/approximation will hold.

The derivative is a numerical value; it is an "instantiation" of the differential ; say f(x)=x , then
the differential df satisfies df=dx (because the graph of f is a line, and the best local-linear approximation to a line is the line itself.) The derivative f'(x) , say at x=1 is f'(1)=1 .




** A student of mine once actually wrote "two samurais" instead of "to summarize" . I don't think it was a joke.

 

[kashan123999]

Thank you everyone for your kind help,now i am beginning to get a lot of intuition of differentials as a change in value of function w.r.t some other variable along the tangent line,when it is drawn...so we usually try to find the change in the value of function w.r.t some variable,to a certain extent right? means we are kind of approaching lim del x --> some value and NOT zero...apologies for i am not a native English speaker,might sound cumbersome,but hope you can give your view to my perplexed understanding...