Is d dt a fraction?
Is dxdt a fraction or not? dxdt is not a fraction it only behaves like a fraction! dx or dt does not have any meaning it is just ddt(x) which has meaning but we treat it as dxdt.
Can differentials divide?
Dividing out by differentials, which is a formal calculation (and thus not rigorous), is not necessary. Although, it is true that, in a sense, you can divide them out (see differential 1-forms). where the f'(x) on the left is the derivative and the dy/dx on the right is the ratio of the differentials.
Can you split dy dx?
Separation of variables only works if we can move the y’s to the left-hand side using multiplication or division, not addition or subtraction. An equation like dy/dx = (x + 3)/(y – 2) is also separable, because we can multiply both sides by (y – 2); it is ok to move constants to either side.
Can you multiply dy dx?
Writing “dy/dx= f(x) so dy= f(x)dx” by “multiplying both sides by dx” is “mathematically unsound” because dy/dx is the limit of a fraction, not a fraction. But you can use that equation: “dy= f(x)dx” (though not the justification “multiplying both sides by dx”) because it is the limit of a fraction!
Is dy a number?
A differential dx is not a real number or variable. Rather, it is a convenient notation in calculus. It can intuitively be thought of as “a very small change in x”, and it makes lots of the notation in calculus seem more sensible.
Why is the derivative of a definite integral always a fraction?
The reason this works out involves the fact that the derivative is defined as the LIMIT of a fraction, and much of its behavior still looks like a fraction; and similarly that the definite integral is defined as the limit of a sum of products with delta x, and its behavior still reflects that (within reason).
Should derivatives be treated as quotients?
I find it redundant that they say “do not …” when you essentially have to do just thatwhen solving many calculus problems by hand. So to reiterate, we know that derivatives are certainly not quotients, but in practice it’s entirely safe to “treat” them as such.
Why is the derivative a limit of a fraction?
The underlying reason, ultimately, is that the derivative is defined as a limit of fractions, and so properties of fractions “survive the passage to the limit”. Here is a question and answer from a couple years ago about your issue:
Should differentials be written as fractions or not?
However, the numerical calculations will remain the same in any case. EDIT. The Leibniz notation ensures that no problem will arise if one treats the differentials as fractions because it beautifully works out in single-variable calculus. But explicitly stating them as ‘fractions’ in any exam/test could cost one the all important marks.