Calculus 220, section 2.2 First and Second Derivative Rules
notes by Tim Pilachowski
Last time, we did a visual review of graphs, looking at six items: increasing/decreasing, maximum/minimum
(relative and absolute), inflection points and concave up/down, x and y-intercepts, points at which the function
is undefined, and asymptotes. This time, we explore how first and second derivatives tell us about some of these
attributes.
Consider the graph of y = x2 pictured to the left along with its derivatives xy 2
=
′
and . The parabola, y = x2=
′′
y2, is decreasing on the interval – ∞ < x < 0, has an
absolute minimum at (0, 0), and is increasing on the interval 0 < x < ∞. The values
of the first derivative, which tells us the slope of the curve, match this behavior.
The first derivative, xy 2
=
′
, is negative on the interval – ∞ < x < 0, equals 0 at
x = 0, and is positive on the interval 0 < x < ∞. The second derivative tells us a
the concavity of y = xbout
2 . The second derivative, 2
=
′
′
ysitive for all values of
x, indicating that the parabola is concave up for all values of x in its domain.
What is th
, is po
e connection between the concavity of a function and its second
st
derivative? The second derivative is the slope of the first derivative, and tells us how the first derivative is
changing, i.e. how the slope of the function is itself changing. In the graph of 2
xy = above, the slope (fir
derivative) is negative on the interval
– ∞ < x < 0. Note that the slope of the parabola is becoming less steep (more shallow) as x approaches 0.
Another way to say the same thing is that the slope of the parabola (first derivative), while still negative, is
becoming less negative as x approaches 0, until the curve hits x = 0, at which point the slope of the parabola
(first derivative) equals 0. On the interval 0 < x < ∞ the slope of the parabola (first derivative) is positive. Note,
too, that the slope of the parabola is becoming steeper (i.e. the first derivative is becoming ever-larger positive
numbers) as x approaches . The slope of the curve = the first derivative is progressing in this way: ∞
very negative < less negative < zero < small positive < large positive
slope of curve is always increasing = first derivative is always increasing
= slope of first derivative is always positive = second derivative is always positive
In trying to describe or draw a curve, we’ll look for critical values, i.e. values at which something significant
happens on the curve, i.e. where the first or second derivative (or both) either equals 0 or is undefined.
Consider 2
1
xxy == . Since. x
x
dx
dy
2
1
2
12
1== − is undefined for x = 0 and
positive for all x > 0, we conclude that the graph of x has a vertical tangent at the
origin and is increasing over the whole domain x ≥ 0. Since 2
3
4
1
2
2−
−= x
dx
yd is
negative for all x > 0 we conclude that the graph has no points of inflection and that
x is concave down over its whole domain.
For . The only critical value is x = 0, for which f and
both its derivatives equal 0. Note however that
xfxfxf 6 and ,3, 23 =
′′
=
′
=
f
′
is positive for all other values of
x. Since f is therefore increasing on both sides of (0, 0), it cannot be either a relative
maximum or minimum. We conclude that the curve “levels out” at the origin then
continues upward and that (0, 0) is a point of inflection. Since 0for 0
<
<
′′ xf the
curve is concave down to the left of the y-axis. Since the curve is
concave up to the right of the y-axis.
0for 0 >>
′′ xf
