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so these are both ways that you will see limit based definitions of derivatives usually this is usually if you're thinking about the derivative at a point here if you're thinking about the derivative in general but these are both equivalent this they're both based on the slope of a tangent line or the instantaneous rate of change and using these I want to establish some of the core properties of derivatives for us and the first one that I'm going to do will seem like common sense or maybe it will once we talk about a little bit so if f of X if our function is equal to a constant value well then f prime of X is going to be equal to zero now why does that make intuitive sense well we could graph it we could graph it so if that's my y-axis that's my x-axis if I wanted to graph y equals f of X it's going to look like that where this is at the value Y is equal to K so this is y is equal to f of X notice no matter what you change X Y does not change the slope of the tangent line here well frankly is the same line it has the slope of 0 no matter how Y is just not changing here and we could use either of these definitions to establish that even further establish it using these these limit definitions so let's let's see the limit as H approaches 0 of f of X plus h well no matter what we input into our function we get K so f of X plus h would be K minus f of X well no matter what we put into that function we get K over H well this is just going to be 0 over H so this limit is just going to be equal to 0 so f prime of X for any for any X the derivative is 0 and you see that here that this is the slope of the tangent line for any X is equal to 0 so someone walks up to you on the street and says ok H of X H of X H of X is equal to PI what is H prime of X well you say well PI it's just a constant value that the value of our function is not changing as we change our X the slope of the tangent line there the instantaneous rate of change it is going to be equal to zero