It means that the first derivative is a constant. The derivative may be with regard to time or any other variable.
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it mean per rate
It means something that doesn't change, like a number.
There is no difference between them they are same rate constant is another name of specific rate constant
accurately means constant accuracy in a problem or equation or expression
Constant just means that the number does not change. For example, in Chemistry, avogadro's number is 6.022x10^23 and that number will never change, therefore it is a constant.