U1S4V04 Derivatives of Linear Functions.txt

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We gave you this function, g(x) equals one half x minus 1.
So this is a line with slope 1/2.
Here's its graph.
And we're thinking about the derivative.
So if we take the derivative, g prime,
just at some random point x, what can we say about that?
Well, if random point x corresponds
to some random point on this graph,
and if we take the tangent line at this place,
then, well, it's pretty much going
to have to run right along this line that we've already drawn.
And we know the slope of this line-- it's one half.
So g prime of any x is going to be one half.
This is telling us that the graph of the derivative,
g prime, is going to be a horizontal line at height one
half, because it takes the value one half at every place.

In general, if g(x) is some random linear function
mx plus b, the same thing is going to hold.
Except instead of one half, we're
going to have that the derivative of g
is always equal to m-- the slope of g.
We can do this algebraically.
We know that g prime of x is defined as a limit.
It's the limit as c approaches x.
So I'm using c here, not b, because I already
have b written in the function.
So limit as c approach as x of g(c) minus g(x) over c minus x.
And if we just plug this into the definition of g,
we get the limit as c approaches x of mc
plus b minus quantity mx plus b over c minus x.
And that's going to be the limit as c approaches x.
And the b's are going to cancel up on top.
And we're left with mc minus mx, which I'm going
to write as m times c minus x.
And our denominator is still c minus x.
And we get more cancellation, and so we just get the limit
as c approaches x of m.
And m is just a constant.
It's our slope of the original line.
So when we take the limit of the constant,
we're just going to get that constant.
So we always get m as the result. So g prime of x
equals m, no matter what the x is.
Now one special case of this that we should point out
is that if g(x) happens to be zero x plus b--
or in other words, just equals b, a constant function-- then
what we just did is telling us that g prime of x equals zero.
So the derivative of a constant function is always zero.
And hopefully that makes sense to you,
because after all, the derivative
is supposed to measure an instantaneous rate of change.
But if a function is constant, then the function
isn't changing at all.
So the derivative of that constant function
should be just zero.

We have some questions for you, and then we'll come back
and we'll look at some slightly more complicated functions.