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Current time:0:00Total duration:2:27

- [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.