To determine acceleration from velocity, you need to take the derivative of the velocity function with respect to time.

Acceleration is defined as the rate at which velocity changes over time. If you have a mathematical function that describes the velocity of an object, you can find the acceleration by differentiating this function with respect to time. Differentiation is a fundamental mathematical process that determines the rate of change of a quantity.

For instance, consider the velocity $v(t)$ of an object described by the function

$v(t) = 3t^2 + 2t + 1.$

To find the acceleration, you will differentiate this velocity function with respect to time. The derivative of $v(t)$ with respect to $t$ is denoted as $\frac{dv}{dt}$.

Applying basic differentiation rules, we differentiate each term of the velocity function as follows:

• The derivative of $3t^2$ is $6t$ (according to the power rule, $\frac{d}{dt}(t^n) = nt^{n-1}$).
• The derivative of $2t$ is $2$ (since $\frac{d}{dt}(t) = 1$).
• The derivative of the constant $1$ is $0$ (as constants do not change).

Thus, we find that

$\frac{dv}{dt} = 6t + 2.$

This resulting function, $6t + 2$, represents the acceleration of the object at any time $t$.

In conclusion, by differentiating the velocity function with respect to time, you derive the acceleration function, which indicates how the velocity of the object changes at any given moment.