Derivative as Velocity
Derivative of position is velocity; of velocity is acceleration.
Key idea
Derivative of position is velocity; of velocity is acceleration.
In depth
If s(t) is position, then s′(t) is velocity and s′′(t) is acceleration. The sign of velocity tells direction; the magnitude is speed. Derivatives translate position into motion.
Worked example
s(t) = t³ − 6t: v(t) = 3t² − 6; at t = 2, v = 6 and the particle moves forward.
Formula
Real-world uses
- Finding speed and acceleration from a position formula in sport science and vehicle testing
Common mistake
Confusing velocity with speed. Speed is |v|.
How to learn this
- Read the key idea and say it in your own words.
- Work through the worked example, then try a similar problem.
- Use the interactive lesson’s visual and mini-quiz to check the pattern.
Frequently asked questions
What is Derivative as Velocity?
Derivative of position is velocity; of velocity is acceleration. It is a topic at the advanced level in Calculus.
What is the formula for Derivative as Velocity?
The key formula is v = ds/dt.
Where is Derivative as Velocity used in real life?
Real-world examples: Finding speed and acceleration from a position formula in sport science and vehicle testing.
What should I learn before Derivative as Velocity?
Make sure you are comfortable with Tangent Lines, the previous topic in Calculus.
What is a common mistake with Derivative as Velocity?
Confusing velocity with speed. Speed is |v|.
How can I practise Derivative as Velocity?
Open the interactive lesson, play with the visual, then answer the three-question mini-quiz. Getting two right marks the topic complete.
What comes after Derivative as Velocity?
Move on to Power Rule next.
Related advanced topics in Calculus
- Idea of a Limit – The value a function approaches as the input approaches a point.
- One-Sided Limits – Left and right limits must agree for the limit to exist.
- Limit Laws – Limits of sums, products and quotients are computed from the parts.
- Factoring and Rationalizing Limits – Remove 0/0 indeterminacy with algebra.
- Limits at Infinity – End behaviour of functions.
- Infinite Limits and Asymptotes – Function values grow without bound near a vertical asymptote.