Let's say the distance for a car equals: $f(x) = 5x$. Then it's speed is $f'(x) = 5$. But what does it mean to have speed of five? So, for example speed of car at $t=10$ is $5$, but what does it mean?
Does it mean that distance will increase in 5 meters in the next second? $f'(x) = f(x+1) - f(x) = f(11) - f(10) = 55 - 50 = 5 == 5$
Or does it mean that distance has increased in 5 meters in the last second? $f'(x) = f(x) - f(x-1) = f(10) - f(9) = 50 - 45 = 5 == 5$
Okay, this explanation might work for a simple function but let's say $f(x) = x^2$ and it won't work anymore.
For example.
$f'(3)=2*3=6$
$f(x+1)-f(x) = f(4) - f(3) = 16 - 9 = 7 <> 6$
$f(x) - f(x-1) = f(3) - f(2) = 9 - 4 = 5 <> 6$
But what I noticed is that for the case $f(x) = x^2$ there is an other correlation: $f'(x) = (f(x+1) + f(x-1))/2 = (f(4) - f(2))/2 = (16-4)/2 = 6 == 6$.
But if we try it with $f(x) = x^3$ then this rule won't work anymore. What will work is $f'(x) - 1 = (f(x+1) + f(x-1))/2$.
For $f(x) = x^4$ it will be $f'(x) - 4*x = (f(x+1) + f(x-1))/2$.
So my questions are:
what derivative mean in terms of $f(x)$? Like if $f'(x) = 5$ then what it can tell me about the change in $f(x)$? And why $f(x+1) <> f(x) + f'(x)$?
whats the reasons behind the correlations i've found for $x^2$, $x^3$, $x^4$?