Expert: Scotto - 6/14/2010

Question
QUESTION: Hi.
So in high school there was no clear definition of differentials but  it has been implied to me  that Differentials are infinitesimal change in a quantity or it's the limit of delta x as it ->0    but when I say that in higher math education they say that I am an idiot or something so what's dx exactly?

ANSWER: The term 'dx' is found in calculus, and most high school teachers didn't even take calculus.
If they did, and they're not teaching mathematics, they've forgotten it.

This may be a long answer, but I hope you read and understand it and really get into it.
It is really a fascinating subject and is the basis for all mathematics.


What goes with dx is an integral.  That is, ∫.
It is found in the form ∫f(x)dx.
This means that integration is to be done on f(x) with respect to x.

Sometimes, in even higher math, you see ∫f(x,y)dx or ∫f(x,y)dy.
The 1st one is a function that is defined over x and y and integrated with respect to x.
The 2nd one is a function that is defined over x and y and integrated with respect to y.

For example, the ∫x^n dx = [x^(n+1)]/(n+1) + C, where C is a constant when there are no limits.
The integral find the area underneath the curve.

For example, it can be seen that f(x)=3x+4 from x=2 to x=4, and the x axis makes a trapezoid.
At x=2, f(2)=10; at x=4, f(4)=16; the average of these two is 13 and the width is 4-2=2.
This means the area is 2*13=26.

To integrate this, we have ∫3x+4 dx = 3x²/2 + 4x; the low point was 2 and the high point was 4.
If the function is f(x), the integral is commonly called F(x).  F(4)=24+16=40; F(2)=6+8=14;
40-14=26, and that was the area given in the last paragraph.


Now back to the difference between the x's getting small, if x² is integrated from 0 to 6,
the answer would be x³/3 evaluated at 6 minus the evaluation 0, so the answer would be 6³/3,
which is 216/3=72.

If we take dx to be 3, there are two intervals.  One has height from 0 to 9 and the other has height from 9 to 36.  Now (0+9)/2=4.5 and (9+36)/2=22.5, and 4.5 + 22.5 = 27.  The width of each interval was 3, so 3*27=81.  This is close to 72, which is the actual area.  If we took 3 intervals, that would be 0 to 2, 2 to 4, and 4 to 6.  The average heights would be (0+4)/2=2,
(4+16)/2=10, and (16+36)/2=26.  Here the width of each interval is 2, so take 2(2+10+26)=2*38=76.
Now 76 is closer to 72 than 81, but not there yet.  It is closer because we used the difference in x as 2 instead of 3.  When the difference is used as dx, the true area is gotten, since then it comes down to integration.

If you're really into calculus, d(sinx)/dx = cosx, so ∫cosx dx = -sinx + C;
d(cosx)/dx = -sinx, so ∫sinx dx = -cosx + C; and d(e^x)/dx = e^x, so ∫e^x dx = e^x + C.

Other functions get messier.  For example, d(tanx)/dx = sec²x + C, d(cscx)/dx = -ctn²x + C,
d(secx)/dx = secx*tanx, and d(cscx) = -cscx*ctnx.

If you really want to get more, note that the derivative of x^n does not require n to be an integer.  For example, take f(x) = √x.  This is the same as f(x) = x^0.5.  The derivative of
f(x), written as f'(x), is f'(x) = 0.5x^(-0.5) = 1/(2√x).  Note that the function is multiplied by the power and then 1 is subtracted from the power.  Since the power was 0.5, the function is multiplied by 0.5 (or divided by 2) and 1 is subtracted from 0.5, so 0.5 - 1 = -0.5.  A negatvie exponet puts the function in the denominator.  Thus, for example, x^-3 = 1/x^3.

I have told you how to differentiate x^n, exponentials, and trig functions.  That covers all the simple functions.  I have also told you how to inegrate x^n, exponentials, and sin and cos.  The other trig functions are a little more challenging to do.  Once all these are known, that is the basis for all mathematics.  To get more advanced, these are combined.

A simple combination is if f(x) = g(x)h(x).  Then the derivative of f(x), f'(x), is defined as
f'(x) = g'(x)h(x) + g(x)h'(x).  For example, take g(x)=x and h(x)=x, then f(x)=x*x=x².
We know that if f(x)=x², f'(x)=2x.  Well, using f(x)=g(x)h(x), we get
f'(x)=g'(x)h(x) + g(x)h'(x).  Note that g'(x) = h'(x) = 1, so f'(x) = x + x = 2x.
In this way we can see that the formula works.

It is similar to 3*8=24.  One you know this, you don't do 8+8+8 all the time, you just know the answer.  In the same way, if f(x) = (x+3)²(x+4)², we know that
f'(x) = 2(x+3)(x+4)² + (x+3)²2(x+4).  Since in the parenthesis is x+3 and x+4, we don't have to worry about the chain rule.

Oh, didn't I mention that?  If f(x) = g(h(x)), then f'(x) = g'(h(x))h'(x).
If f(x) = (x+3)², f'(x) = 2(x+3)d(x+3)/dx, d(x+3)/dx = 1.

If this thrills you, I can answer more questions.
Like how it is used in physics - the derivative of distance is speed, and the derivative of speet is acceleration.


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QUESTION: Thanks for the information.
But sorry I guess my question wasn't very clear I do know calculus and differentiation and integration and that f'(x) = limit h->0 (f(x+h)-f(x))/h and rates of changes and that stuff
and integral is sum of the infinitesimal small rectangles with length of f(x) and width of dx what I was trying to ask about differential forms like this http://en.wikipedia.org/wiki/Differential_%28infinitesimal%29 and this http://en.wikipedia.org/wiki/Differential_form this things they never taught us in school

Answer
I put what was given in the search box.

I found  http://en.wikipedia.org/wiki/Differential, but not the first.

It transmits as, "http://en.wikipedia.org/wiki/Differential_%28infinitesimal%29",
and for that, I found the 2nd one.

One way to use them is when something changes over time in a linear fashion over a period of, say, days, then the differential to use would be the daily difference.  This could be computed from the average of however many days in the past were used.  Another way of doing it would be to use numerical analysis to fit a line (or appropriate curve) to the data and actually computing the derivative.  Using differentials avoids this work.

I'm not sure exactly what your question is.

As far as the use, I didn't see a whole lot of use of differentials until I took some senior level math courses and the graduate math courses.