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# Derivative & the direction of a function

## Video transcript

A function f of x
is plotted below. Highlight an interval
where f prime of x with the first derivative
of f with respect to x is greater than 0. So if our derivative, f prime
of x, is greater than 0, that means that the slope of
the tangent line is positive. That means that
the function must be increasing
whenever this is true. So where is the
function increasing? So over here we see
that as x increases, our function is going to
smaller and smaller values. So our function is
decreasing this entire place. Then it's decreasing at
slower and slower rates all the way to the
point right over here where the slope of the
tangent line is flat. And then the function
starts increasing. And it starts increasing at
faster and faster and faster rates. So we could put this blue
area anywhere from here to anywhere in this region
here where the function itself is increasing. Notice, the function
could even be negative, but the function is increasing. So even this would
be a valid place. That would be a valid place. Anywhere where the function's
rate of-- when x increases, the function is
actually getting larger. So let's check our answer. And let's do a
couple more of these. A function f of x
is plotted below. Highlight an interval where
f of x is greater than 0. So f of x should be sitting in
the first or second quadrant right over here. It needs to be greater
than 0, and f prime of x is less than 0. So f prime of x
less than 0 means that the function is decreasing. The slope of the tangent
line is negative. So let's think about it. The two areas where f
of x is greater than 0 are this interval
right over here and this interval
right over here. But we also care about
the function decreasing. This won't be valid, because
the function is increasing here. It's increasing at
slower and slower rates, but it is increasing. The function is going
up as x goes up. If we go over here, the function
is decreasing as x increases. So this seems to
meet our constraints. This area right over here
won't because the function is positive, but
it's also increasing. So here the derivative
is positive. Here the derivative is positive. Here the derivative is negative. So the function
itself is positive, but the slope of the
tangent line is negative. It's decreasing as x increases. Let's do one more. The function f is plotted below. Highlight an interval where
f of x is greater than 0 and f prime of x is less than 0. So the same exact idea--
where are we positive but our function is
actually decreasing? So it's positive here,
but we can't pick that. It's positive here,
but we can't pick that. The function is decreasing here,
but the function isn't positive there. The derivative is negative, but
the function isn't positive. So that wouldn't make it. So this is the only region
that we can throw it in. And we got it right again.