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- [Voiceover] So these are both ways that you will see limit-based definitions of derivatives. 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. They're both based on the slope of a tangent line, or the instantaneous rate of change, and using these, I wanna 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 it 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 gonna 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 a slope of zero. 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 limit definitions, so let's see, the limit as h approaches zero 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 zero over h so this limit is just going to be equal to zero. So f prime of x for any, for any x, the derivative is zero, and you see that here, that the slope of the tangent line for any x is equal to zero. So if someone walks up to you on the street and says, "Okay, h of x, h of x, h of x, is equal to pi, "what is h prime of x?" Well, you say, well, pi, that'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.