Why do derivatives of sin(2πft) include the 2πf coefficient?

[Joao]

Hi everyone! I'm a psychologist form Brazil, so sorry for the bad English and for the lack of knowledge in math!

I ve been trying to understand the Schrödinger equation and, as predicted, it's very hard!

Please, help me with this:

A sine wave function can be written as:

F (x) = sin (x)

And that can be represented as psi.

The the derivative of that function can be written as:

F (x) = cos (x)

And that can be represented as psi dot.

Also, the derivative of psi dot can be written as:

F (x) = - sin (x)

And that's psi with two dots.

So far I understood!

But things get confusing here:

A wave equation can be written as:

F (x) = A sin (2π f x )

Where A stands for the amplitude, 2π f x stands for the period.

But, in this video:



It is stated, at 8 mins, that the correct wave function is, as expected:

F (x)= sin (2π f t)

And that's represented by psi

But the derivative of that is

F (x) = 2π f cos (2π f t)

And not

F (x) = cos (2π f t)

(And that's represented by psi dot)

To make things worst, the derivative of that last equation, psi dot, is represented by

$f (x) = - (2π f)^2 sin (2π f t)$

And not just

F (x) = - sin (2π f t)

(And that's represented by psi with 2 dots)

Why is that?

Thanks a lot!

 

[PeroK]

@Joao you need a calculus course.

Dots above a function normally denote a time derivative. The normal convention for a derivative is either $f'$ or $\frac{df}{dx}$.

Then, of course, you have "partial" derivatives $\frac{\partial \psi}{\partial x}$ and $\frac{\partial \psi}{\partial t}$ etc.

In answer to your question, there is a little thing called the chain rule, which is quite important!

 

[PeroK]

@Joao try:

http://tutorial.math.lamar.edu/

If you want to learn about calculus.

 

[Joao]

Thanks a lot! I will do that! So, that makes sense, the derivative of

Ψ = Sin (2πft)
Is
Ψ(dot) = 2πf cos (2πft)
And not
Ψ(dot) = cos (2πft)

And after I make the calculus course, I will understand why?

Really sorry to bother!

 

[PeroK]

Thanks a lot! I will do that! So, that makes sense, the derivative of

Ψ = Sin (2πft)
Is
Ψ(dot) = 2πf cos (2πft)
And not
Ψ(dot) = cos (2πft)

And after I make the calculus course, I will understand why?

Really sorry to bother!

Yes. You could, say, draw a graph of $\sin(2x)$ and look at the slope. You'll see that it is steeper than the graph of $\sin(x)$.

In fact, it's twice as steep at each point $x$.

Which is what you get from the chain rule.

 

[Joao]

Yes. You could, say, draw a graph of $\sin(2x)$ and look at the slope. You'll see that it is steeper than the graph of $\sin(x)$.

In fact, it's twice as steep at each point $x$.

Which is what you get from the chain rule.

Thanks a lot for your time! I'll do the calculus course! =)))))

Happy 2018!