A derivative finds the rate of change of a function at
any given point. Before discussing the rate of change, let’s understand the
concept of change rate. The rate of change represents the amount of change,
including average change rates and instantaneous change rates.
Average Rate of Change vs.
Instantaneous Rate of Change
Consider a function f(x) that represents data
distributed similarly to the above diagram. 'f' stands for function, and 'x' is
the variable used as input. For example, if f(x) = 2x2 + x + 3,
increasing the value of 'x' and calculating the function (y) values would yield
a similar graph as shown above.
Now, if the value of 'x' changes from 'a' to 'b', how
much does the function output (y) change in proportion to 'x'? This change is
found using the average rate of change. The symbol Δ (delta) used in the
formula represents the change amount, where Δy indicates the change in 'y' and
Δx indicates the change in 'x'. The average rate of change is obtained by
dividing these two values.
If we consider a car traveling over time and create a
function from the data of the car's distance, with 'x' as time and 'y' as the
distance traveled, the average rate of change between time 'a' and time 'b'
would represent the car's speed.
The concept of instantaneous rate of change, which
means differentiation, is a bit more challenging compared to the average rate
of change. While average rate of change is calculated over a clearly
distinguishable interval (from 'a' to 'b'), instantaneous rate of change
calculates the rate of change at a single point.
To make it easier, let’s use Δx to indicate the change
in 'x'. In the average rate of change, Δx is relatively large, but for the
instantaneous rate of change, Δx is extremely small. As Δx becomes close to
zero (lim: limit), the value of the instantaneous rate of change can be found.
For example, speed cameras measure the speed of a car
as it passes a certain point, not between two points ('a' and 'b'), but at a
specific location. In this case, differentiation is used. By making the
measurement range close to zero, the speed can be calculated at a particular
point.
Differentiation Formula
There may be times when we want to know the result of
differentiating at any point, not just a specific point. In such cases,
differentiating the function and expressing the differentiation result as
another function would be useful. While it’s too lengthy to explain the
principle here, only the basic formulas are provided. There are various
differentiation formulas for different functions, but only the ones used in
reinforcement learning are summarized above. Since this is not a mathematics
test, use the formulas as reference whenever needed.
