The formula for calculating physical damage received is as follow:

Damage Received = Damage dealt by enemy * 120/(120 * Physical Defense)

Lets use some symbols.

Let y = Damage Received

x = Physical Defense

And let damage dealt by enemy = 100.

Therefore, the above equation becomes**y = 100 * 120/(120+x)**

Below, we have the graph of the above equation and some sample values of x and y in the table

From the table above we see that at x=50, ie. defense 50 we will receive 70.5 damage if enemy does 100 damage. So we receive only around 70% of the total damage dealt when our physical defense is 50

At x=120, ie defense 120, we receive exactly half (50%) of the damage dealt.

We can see here that increasing x gives us diminishing returns

So as you have more armor, buying even more armor results in lesser returns.

## Finding the rate of change (derivative) of the equation

We are interested to see how damage received (y) changes as we increase armor (x)

Derivatives will give us that information

Our equation is :

y = 100 * 120/(120+x)

By using quotient rule we find its derivative which is

y’ = -100*120/(120+x)^{2}

The -ve sign indicates that y (damage) decreases when x (armor) increases

For simplicity let us just write the eq without the -ve sign**y’ = 100*120/(120+x) ^{2} **

The above equation tells us how y changes with respect to x.

The table of sample values and graph of the equation y’ = 100*120/(120+x)^{2}** ** is below

From the table, we see that the rate of change of y at x=50 is 0.41 ie.,

at x (physical defense) = 50 each unit increase in x will give us 0.41 less damage (when 100 damage is dealt) . And at 120 physical defense each unit increase in physical defense will give us only 0.20 less dmg.

Thus we can easily see that at 120 physical defense buying more armor is half as efficient as buying armor at 50. And at 200 physical defense buying more armor is around quarter less efficient as buying armor at 50

The derivative gives us a clear picture how increasing physical defense/armor has diminishing returns

Graphs drawn using https://www.desmos.com/calculator

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