How to prove the derivative of position is velocity and of velocity is acceleration?
How has it been proven that the derivative of position is velocity and the derivative of velocity is acceleration? From Google searching, it seems that everyone just states it as fact without any proof behind it.
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$\begingroup$Velocity is defined as the rate of change of displacement with respect to time, and acceleration is defined as the rate of change of velocity with respect to time.
$\endgroup$ $\begingroup$If we define change in position as $\delta x$ and a change in time as $\delta t$, then we, know that the average velocity over that time is given by $$\frac{\delta x}{\delta t}.$$ But, we do not want to know the average velocity, so we want to find the instantaneous velocity. In this case, we, by the definition of the derivative, want to take the limit as $\delta t\rightarrow0$. So, we have as velocity $$\lim_{\delta t \to 0} \frac{\delta x}{\delta t}=x'(t)$$ as desired. This isn't really a proof, per se, but it is a deeper explanation which perhaps you are seeking. A similar explanation applies to acceleration.
$\endgroup$ $\begingroup$The derivative is the slope of the function. So if the function is $f(x)=5x-3$, then $f'(x)=5$, because the derivative is the slope of the function. Velocity is the change in position, so it's the slope of the position. Acceleration is the change in velocity, so it is the change in velocity. Since derivatives are about slope, that is how the derivative of position is velocity, and the derivative of velocity is acceleration. So if the position can be expressed with the function $f(x)=x^2 - 3x + 7$, then the derivative would be $f'(x)=2x-3$ since that is the slope of the function at any given point, and since it is the slope of the position function, it is velocity. Same for acceleration; $f"(x)=2$, which is the derivative of velocity, which makes it slope. The slope of velocity is acceleration. This is how the derivative of position is velocity and the derivative is position.
NOTE: These functions are entirely hypothetical and were created on the spur of the moment.
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