I Function of function derivative

1. Feb 17, 2017

rabbed

If p is a function of x which is a function of t and you evaluate delta_p/delta_t as
delta_t goes to zero, it should be possible that delta_p/delta_t equals delta_p/dx
(or dp/dx) before reaching dp/dt.
Is it possible to find an expression for t where this happens?

Hm.. maybe when t = x^-1(dx) ?
Is it possible to find dp/dt for that t?

Last edited: Feb 17, 2017
2. Feb 17, 2017

Simon Bridge

$$\frac{dp}{dt} = \frac{dp}{dx}\frac{dx}{dt}= \lim_{\Delta t \to 0} \frac{p(t+\Delta t) - p(t)}{\Delta t}$$

3. Feb 17, 2017

rabbed

I know, but that doesn't get me an expression of dp(x^-1(dx))/dt that can be evaluated at any t, does it?

On second thought, it should be dp(t+x^-1(dx))/dt

I'm trying to follow what happens to a 2D vector derivative when it starts to grow orthogonal to the tangent..

Last edited: Feb 17, 2017
4. Feb 17, 2017

rabbed

Possible to turn this limit expression into derivatives?

Lim delta_x-> 0: ( p(x^-1(x+delta_x)) - p(x^-1(x)) ) / x^-1(delta_x)

Where x^-1(x) = t

Last edited: Feb 17, 2017
5. Feb 17, 2017

Stephen Tashi

You appear to be using the intuitive idea that a "limit" involves the notion of something "approaching" something else over an interval of time or in a step-by-step fashion. If you look at the formal definition of "$\lim_{t \rightarrow a} f(t)$ you will find that the definition does not define any process taking place in time or in a sequence of steps. Since the definition of a derivative is based on the definition of limit, the definition of derivative also does not involve a process of something "approaching" something else as time passes or as a number of steps are executed. So your question doesn't have any defined meaning in mathematics, because there is no process described in the definition of derivative that would involve a "before" or "after".

If you are talking about algorithms to approximate derivatives, these often do involve a specific sequence of steps. But in order to determine if a variable in such an algorithm "reaches" a certain value before another value, you would have to say which particular algorithm you are asking about.

6. Feb 17, 2017

rabbed

Hi Stephen

I'm trying to increase understanding of what happens to a parametric 2D vector when you take its derivative.
Letting two points of a curve approach eachother by letting the parameter difference go to zero, there should
be a point where the derivative has a direction normal to the curve but has length 0, and then the length should
start to grow, still having the same direction.
I'm thinking that maybe the zero-length point occurs when delta_t = x^-1(dx), and as you decrease delta_t down
to dt the length starts to grow. It should make some sense, since the zero-length point should exist?

7. Feb 17, 2017

rabbed

Wouldn't this take you closer to that idea?

Lim delta_x-> 0: ( p(x^-1(x+delta_x)) - p(x^-1(x)) ) / x^-1(delta_x)

Seems it's called calculus of variations, if you derivate wrt a function?

8. Feb 17, 2017

Stephen Tashi

You apparently are thinking of some algorithm or process to approximate the derivative because, as I mentioned, "taking" a derivative is not defined in terms of process that takes place in time or in a series of steps.

Your original post didn't mention a vector. Apparently you mean a function $F(x) = (f_1(x), f_2(x))$ whose domain is a set of real numbers and whose codomain is a set of two dimensional vectors?

Which derivative? $(f_1'(x), f_2'(x))$?

Why do you think that? Suppose the curve is $F(x) = (f_1(x), f_2(x)) = (x, x+1)$ with $x(t) = t$. Where is there a point point on the curve where $(f_1'(x),f_2'(x))$ is normal to the curve?

9. Feb 18, 2017

rabbed

I know, sorry. But this would apply to each partial derivative so I thought it would be simpler to discuss for just one variable.

I'm thinking of a picture like the one in the answer here: http://math.stackexchange.com/quest...e-of-a-vector-orthogonal-to-the-vector-itself
delta_v would at some point become a zero vector, before starting to grow? And since the vector derivative is created by derivating each component, it should apply to a function of a function of a single variable also?

10. Feb 18, 2017

Stephen Tashi

11. Feb 18, 2017

rabbed

Since I want to study this per component

12. Feb 18, 2017

Stephen Tashi

Then it isn't clear what you are asking. If you can't find the words to express your general question, try asking about a specific example.

13. Feb 18, 2017