Difference between d/dx and dv/dx

Hi

I am looking at the M3 book, page 15, on acceleration depending on displacement. So here is what they write...

a = dv/dt = dv/dx * dx/dt

since dx/dt = v

this gives a = v * dv/dx

Since d/dx (v^2) = 2v * dv/dx

a = d/dx * 0.5v^2

That I don't get is what the difference between d/dx and dv/dx is... I don't understand the "syntax" of it...

If someone could explain, and also go through this in more detail especially where they go from a = v * dv/dx to a = d/dx * 0.5v^2 I would be very greatful.

Thank You

Michail

Reply 1

topcrabcakes

Since d/dx = v^2 = 2v * dv/dx

a = d/dx * 0.5v^2

This should read:

"since d/dx(v^2) = 2v * dv/dx"

so:

a = v * dv/dx = (1/2) d/dx(v^2) = d/dx(0.5*v^2)

Reply 2

MichailZ

Yes, sorry, easy to do this like that, that's what it says in the book.

What I don't get is what the "syntax" of it is... like if I said

S(1-2) (well the integral sign with the 2 and 1 written on it I would know that it means integrate this between these limits so find f(x) then f(a) - f(b)...

and if it just said S(y)dx (I would know it means integrate y with respect to x (i.e. just symbolic, then + c)

but what is the difference between d/dx and dy/dx? (or d/dx and dv/dx)

:frown:

Michail

Reply 3

Rabite

d/dx(f(x)) means "If y were f(x), dy/dx would be..."

So like.
d/dx(x²) = 2x.

Reply 4

MichailZ

I see....! Thanks very much... so d/dx is not really a part of an equation, just an "operation" like a minus sign or something?

Like if I said y * dy/dx it would mean whatever f(x) was at the start multiplied by it's derivative (or something like that) but the dy/dx would actually take part in the overall equation... as opposed to d/dx being an order to differentiate something.... like a minus sign is an order to take something away...

Thanks very much this makes it much clearer!

Michail

Reply 5

BovineBeast

Rabite

d/dx(f(x)) means "If y were f(x), dy/dx would be..."

So like.
d/dx(x²) = 2x.

I think it'd be better to write that (d/dx)x^2.

Reply 6

AlphaNumeric

dv/dx is "The derivative with respect to x of v"

d/dx is what is known as a differential operator. It's an operator since you operate on things with it. It is not a set thing in the sense that dv/dx is, but it tells you do so something to whatever you apply it to, that being "take the derivative with respect to x".

If you apply it to v you get dv/dx
If you apply it to v^2 you get d(v^2)/dx = 2v.dv/dx
If you apply it to yv you get y.dv/dx + v.dy/dx

:smile: