Is derivative a good approximation?

Question

Is derivative an approximate number when x->0 so y tends to a number ? for example we define u=x'(t) and imagine we have x'(t1)=5 does this mean that velocity is very very near to 5 or is it exactly 5 ?

Answer 1

As @littleO said in the comment, if $x'(t)=5$, this means that velocity at time t is exactly 5, not close or very very near to 5.

In motion average velocity is defined as ratio of total distance to total time taken. That is,

$<Velocity>$ = Distance Travelled/Total Time.

But that is average velocity, not instantaneous, it is average velocity, as we can see that, we just took mean of distance over the total time.

Instantaneous velocity is velocity at the instant and can not be defined with just values, it needs relation between position and time, to be exact.

Hence it is defined as:

$V_{instantaneous}$ = dx/dt

Here we can also know that slope of the position and time graph at any time t, represents the velocity dv at that very time.