Question

My textbook just told me that m(dv/dt) can be rewritten as d/dt(mv). My question is, what does the expression d/dt represent? I see these dx's and dy's and dt's all over the place and kind of understand, but I think I'm missing something fundamental. Anyone care to enlighten me?

Answer 1

d/dx is a notation or a mathematical symbol to denote a derivative. dt/dx implies that the derivative of the function t with respect to x. If t = f(x) = x^2 -25 is a function; t denotes the dependent variable and x denotes the independent variable. dt/dx = 2x in this equation

Answer 2

So what does the phrase "with respect to" mean? Just a ratio?

Answer 3

wrt a variable, t for time or s for distance or x or whatever....

Answer 4

the word "respect" particularly, is that the same as saying "compared to"?

Answer 5

y = f(x) = 2x , say x is the variableand we wish to differentiate the function (f(x)) with respect to x then d(f(x)/dx = 2

Answer 6

You could have a function, f(x,y) = 3x +4y and if I say differentiate that, it doesn't make any sense unless I stipulate with respect to what variable.

Answer 7

So if you said differentiate f(x,y) = 3x + 4y with respect to x, you are really asking "how much does a change in x affect the change in f(x,y)"?

Answer 8

Usually you specify one or more variable as independant and some other as dependant o(on the variables)

Answer 9

Ok, I assume that's stuff I will learn about in the next calc class :) I think I get it though, all that original equation I put was saying is \[\sum_{}^{}F = \frac{ d }{ dt }mv\] Which really just means the sum of the forces is equivalent to the mass times the change in velocity, or in other words F=ma, right?

Answer 10

Yes, rate of change of momentum in that (usually equivalent) version...

Answer 11

ok, thanks!