Question

Could someone explain what dx/dt mean.

I understand that d/dx means the derivative with respect to x and I assume dx/dt means the derivative of x with respect to t, which to my understanding is implicit differentiation but can it have other meaning?

Answer 1

In physics, we use dx/dt where x is the is a function of its displacement, and t is time, to show that the derivative of that function is its velocity.

It might sometimes be used as a substitution method when you deal with integrals, but usually you use dx/du where u is the substitution.

Answer 2

If you have a formula in the form of:

\[x=f(t)\]
Then the meaning of \(\frac{dx}{dt}\) is the instantaneous change of \(t\) with respect to \(x\)

Answer 3

No. You are correct. d\dt is just an operator that represents the derivative with respect to the independent variable, t. If you have multiple independent variables, say x and y, then you can have other operators d/dx or d/dy in the same equation. Then these become partial derivatives, acknowledging the fact that there are more independent variables involved. when you have functions being functions of independent variables, then you use implicit differentiation, which is what you noted. But you have the idea correct, it means the same thing in all cases.

Answer 4

How do I use it in U substitution?

Answer 5

Its not something that you need to know yet probably, so I don't want to confuse you too much. Basically, when you are integrating a function, they can look like nested functions. here's an example:\[f(x)=(x^2+5)^3\]
If you let x^2 + 5 be g(x)
and let x^3 by h(x)
f(x) is simply h(g(x))
In this example, you would let x^2+5 be u, because it is easier to integrate u^3. There is more to it, but that gives you an idea why we use it.

Answer 6

I would like to know the rest. So far I am with you.

Answer 7

@mathis1