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# Matching functions & their derivatives graphically

Given the graph of a function, we are asked to recognize the graph of its derivative. Created by Sal Khan.

Video transcript

I have a function
f of x here, and I want to think about
which of these curves could represent f prime
of x, could represent the derivative of f of x. Well, to think
about that, we just have to think
about, well, what is a slope of the tangent line
doing at each point of f of x and see if this
corresponds to that slope, if the value of these functions
correspond to that slope. So we can see when x
is equal to negative 4, the slope of the tangent
line is essentially vertical. So you could say it's
not really defined there. But as we go slightly to the
right of x equals negative 4, we just have a very,
very, very positive slope. So you could kind of
view it as our slope is going from infinity to very,
very positive to a little bit less positive to
a little bit less positive, to a little
bit less positive, to a little bit less positive. So which of these graphs
here have that property? Remember, this is trying
to graph the slope. So which of these functions down
here, which of these graphs, have a value that is essentially
kind of approaching infinity when x is equal to
negative 4, and then it gets less and less and less
positive as x goes to 0? So this one, it looks like it's
coming from negative infinity, and it's getting less and
less and less negative. So that doesn't seem to
meet our constraints. This one looks like it is
coming from positive infinity, and it's getting less and
less and less positive, so that seems to be OK. This has the same property. It's getting less and
less and less positive. This one right over here
starts very negative and gets less and less
and less negative. So we can rule that out. Now let's think about what
happens when x gets to 0. When x gets to 0, the
tangent line is horizontal. We're at a maximum point of
this curve right over here. The slope of a
horizontal line is 0. Remember, we're trying to look
for which one of these curves represent the value
of that slope. So which one of these curves
hit 0 when x is equal to 0? Well, this one doesn't. So the only candidate that
we have left is this one, and this one does hit
0 when x equals 0. And let's see if
it keeps satisfying what we need for f prime of x. So after that point,
it should start getting more and more negative. The slope should get more
and more and more negative, essentially approaching negative
infinity as x approaches 4. And we see that here. The value of this function is
getting more and more negative, and it's approaching negative
infinity as x approaches 4. So we'll go with this one. This looks like a pretty good
candidate for f prime of x.