Question

What is computing derivative definition and how do you use `f(x+h)-f(x)/h`?

Answer 1

Computing the derivative (of a function at a certain point) is calculating the rate at which the function's value is changing at that point.

In the graphs below we can see how a function's value is changing between a point `x` and a point `x+h`

The green line indicates the rate at which the function is changing as a ratio of the change in the function value divided by the difference in width between the two point locations.
This ratio is:
`(f(x+h) - f(x))/h`

As the width, `h` becomes smaller the rate of change line (called the "slope") becomes closer to a theoretical rate of change for the function at a single point.

This theoretical rate of change at a single point is called the derivative of the function at that point.

It can be written as
`lim_(h rarr 0) ( f(x+h) - f(x) ) /(h)`

Obviously there would be problems of division by zero if `h` were actually to reach `0`; but we can reduce it to a very small amount with no issues.

As an example of how this might be applied algebraically, suppose
`f(x) = 3x^2`
then the derivative of f(x), written as
`(d f(x))/(dx)` (the ratio of the change in f for a change in x)
and
`(d f(x))/(dx) = lim_(h rarr 0) ( f(x+h) - f(x) ) /(h)`

`= lim_(h rarr 0) (3(x+h)^2 - 3x^2)/h`

`= lim_(h rarr 0) (3x^2 + 6xh +h^2 - 3x^2)/h`

`= lim_(h rarr 0) ( 6xh +h^2 )/h`

`= lim_(h rarr 0) ( 6x +h )` provided `h` is not exactly `0`

`= 6x`

So we have now calculated the derivative of `f(x) = 3x^2` as a general expression.

If we need to know the derivative (rate of change) of `f(x)` at a particular point, say `x=2`, we simply insert that value for `x` into the general expression we developed for the derivative, namely
`(d f(2))/(dx) = 6(2) = 12`