Derivative of infinitesimal value

[mertcan]

Hi, it may be interesting question but what is the
d (dy)/dx (y is function of x)? I think it is nearly zero but if it is 0 then how can we prove it?

 

[fresh_42]

Hi, it may be interesting question but what is the
d (dy)/dx (y is function of x)? I think it is nearly zero but if it is 0 then how can we prove it?

What do you mean, resp. do you have an example? If you mean $\dfrac{d}{dx}\dfrac{d}{dx} y(x) = \dfrac{d^2 y(x)}{dx^2}$ then it is simply the second derivative, the curvature. If you mean $\dfrac{d}{dx} dy$ then $dy$ is a linear function and $\dfrac{d}{dx} dy = dy$.

 

[mertcan]

What do you mean, resp. do you have an example? If you mean $\dfrac{d}{dx}\dfrac{d}{dx} y(x) = \dfrac{d^2 y(x)}{dx^2}$ then it is simply the second derivative, the curvature. If you mean $\dfrac{d}{dx} dy$ then $dy$ is a linear function and $\dfrac{d}{dx} dy = dy$.

I mean the second one but why the result is dy?

 

[fresh_42]

I mean the second one but why the result is dy?

Because the tangent of a tangent is the tangent again.

 

[Stephen Tashi]

I think it is nearly zero but if it is 0 then how can we prove it?

If you are taking about higher derivatives they need not be functions that are indentically zero. For example let $y = x^3$ and interpret d ( dy/dx) /dx to be the second derivative of $f(x)$.

I you are reasoning with "infinitesimals", then it will be difficult to ask a precise questions. ( Intuitively, all "infinitesimals" are nearly zero !)

If are using differential notation like "dy" to denote the sophisticated idea of a mapping in differential geometry then @fresh_42 has answered your question.