Instantaneous Velocity and Acceleration

The instantaneous velocity of an object is given as the slope of the tangent line to the position-time curve.

$$\vec{v} = \lim_{ \Delta t \to 0 } \frac{\Delta\vec{x}}{\Delta t} = \frac{d\vec{x}}{dt}$$

The instantaneous acceleration is the slope of the line that is tangent to the velocity-time curve.

$$\vec{a} = \lim_{ \Delta t \to 0 } \frac{\Delta\vec{v}}{\Delta t} = \frac{d\vec{v}}{dt}$$

$$\vec{a} = \frac{d \vec{v}}{dt} = \frac{d^2 \vec{x}}{dt^2}$$