Hello, I was pondering acceleration's relation to velocity, when I came across a problem that is more of an issue of differentiation.(adsbygoogle = window.adsbygoogle || []).push({});

Given our variables

v=velocity

a=acceleration

α=slope of 'a'

t=time

(a_{t1→t2})=Total acceleration from t_{1}to t_{2}

(α_{t1→t2})=Total slope of 'a' from t_{1}to t_{2}

[itex]a=\frac{dv}{dt}[/itex]

[itex]α=\frac{da}{dt}[/itex]

would the two equations above imply that: [itex]α=\frac{{d^2}v}{d{t^2}}[/itex]

Since I can find (assuming a has no slope) v(t) with: [itex]a_{{t_1}→{t_2}}=\int_{t_1}^{t_2} a\;dt[/itex]

we can apply this to the slope of 'a' that:

[tex]α_{{t_1}→{t_2}}=\int_{t_1}^{t_2} α\;dt[/tex]

would this also imply that the total slope of 'v' with reguard to the slope of 'a' from t_{1}to t_{2}be equal to:

[tex]\int\int_{t_1}^{t_2} α\; dt[/tex]

I ask this in the differentiation page since it seems this applies generally to calculus. Converting to generall variables (v=y t=x) would:

[tex]f(x)=\int\int \frac{{d^2}y}{d{x^2}}\;dx[/tex]

I'm asking to clairify. I also noticed that if you look at this algebriaclly you get (given that [itex]α=\frac{{d^2}y}{d{x^2}}[/itex])

[tex]α=\frac{y}{x^2}\Rightarrow y=α{x^2}[/tex]

but that gives a different answer then the integration, unless if I square x (which I only misunderstand because of the fact, that Leibnez's notation squares the 'd' instead of the 'y'). When I do square 'y', I get the same answer as when I integrate.

so basiclly what im asking is:

to find the value of f(x), can i integrate all the way from whatever slope of f(x)?

can i express this algebriaclly by putting y to the power of the [number of slopes], and x to the same power respectivly?

if [itex]α=\frac{dy}{dx}[/itex] and if [itex]β=\frac{dα}{dx}[/itex], does that automaticly mean that [itex]β=\frac{{d^2}y}{d{x^2}}[/itex]

?

sorry if i made any silly asumptions in my math, im new to calculus and math-interpreted-physics, thank you.

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# Second, Third (ect) Derivatives, and their relation to the function value

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