When we want to find the velocity of an object we use the derivative to find this. However, I just learned that when you find the acceleration of the object you find the second derivative.
I'm confused on what is being defined as the parameters of acceleration. I always thought acceleration of an object is it's velocity (d/t).
Furthermore, in the second derivative are we using the x value or the y value of interest. In the first derivative we were only concerned with the x value. Does this still hold true with the second derivative?
I would post pictures but apparently I'm still lacking 4 points.
Best Answer
The velocity is the rate of change of displacement.
The acceleration is the rate of change of velocity.
So the velocity is the derivative with respect to $t$ of the displacement function $s(t)$. In symbols, $v(t)=s'(t)$.
The acceleration is the derivative of velocity with respect to $t$. In symbols, $a(t)=v'(t)$.
It follows that $a(t)$ is the second derivative of displacement. In symbols, $a(t)=s''(t)$.
If you prefer Leibniz notation, let $s$ be displacement at time $t$. Then the velocity is $\dfrac{ds}{dt}$ and the acceleration is $\dfrac{d}{dt}\left(\dfrac{ds}{dt}\right)$, which is $\dfrac{d^2s}{dt^2}$.