Introduction to Derivative
Introduction to
Derivative
Instantaneous velocity
If we drive an automobile from one town to another 80 miles away in 2 hours, our average velocity is 40
miles per hour. But during our trip the speedometer reading was often different from 40.At
the start, it registered 0; at times it rose as high as 57. What the speedometer reads at a
particular instant is the instantaneous velocity at that instant.
Average velocity is counted by dividing the distance traveled by the time taken to travel that distance.
The smaller the time interval, the closer the average velocity over that interval is to the instantaneous velocity.
If we let the time interval shrink to zero, the average velocity approaches the instantaneous velocity.
\[
v = \lim_{Δt \to 0} v_{avg} = \lim_{Δt \to 0} \frac{f(x+Δt) - f(x)}{Δt}
\]
Definition of a Derivative
Using the example of velocity above, derivative of a function at a point is the limit of the average rate of change of the function over an interval as the interval shrinks to zero.
\[
f'(a) = \lim_{h \to 0} \frac{f(a+h) - f(a)}{h}
\]
