Refer to figure.
V_X is the speed at which an aeroplane achieves its maximum angle or gradient of climb.
Since angle of climb depends on excess thrust, V_X is the speed at which the difference between thrust available and drag is greatest.

Because thrust available against speed is different for turbojet and propeller aeroplanes, it must be established at what speed V_X is for both types of aeroplanes.

For best angle of climb we need maximum excess thrust.
If thrust were constant at all airspeeds, then at V_MD where drag is minimum, the excess thrust and climb angle would be maximum.

For maximum endurance we must fly at the speed at which fuel flow is minimum.

Rate of climb is maximum when excess power is maximum.
This occurs at approximately V_MP.
In prop aircraft fuel flow is proportional to power, so minimum fuel flow occurs at the speed at which power required is minimum.
This speed is V_MP.

Rate of climb is equal to excess power divided by aircraft weight.