that the gas volume is expanding at a rate of 2inch3 per
minute, then what is the rate of change of the pressure?
(c) the slope of the tangent to the parabola at P ,
(d) the angle \OP Q where Q is the point (0, 3).
(b) The ideal gas law turns out to be on
Using derivatives to approximate numbers.
(a) Find the derivative of f (x) = x4/3 .
(b) Use (a) to estimate the number
127
4/ 3
(b) Show that for any pair of functions u and v one
has
(uv )0
u0
v0
=
+
uv
u
v
(u/v )0
u0
v0
=
u/v
u
v
4/ 3
125
2
approximatel
A depends on B depends on C depends on. . .
Someone is pumping water into a balloon. Assuming
that the balloon is spherical you can say how large it
is by specifying its radius R. For a growing balloon
this radius will change with time t.
The volume of th
13.4. Example where you really need the Chain Rule. We know what the derivative of sin x with
respect to x is, but none of the rules we have found so far tell us how to dierentiate f (x) = sin(2x).
The function f (x) = sin 2x is the composition of two sim
The direct approach goes like this:
0
f ( x) =
d1
=
1
4
=
1
4
x4
dx
1
4
x
x4 )
3/4 d(1
x4
1
1 /4
dx
3 /4
4x3
x3
=
1
x4
3/4
To nd the derivative using implicit dierentiation we must rst nd a nice implicit description of the
function. For instance, we could
For example, if y =
p
1
p
, then y = 1/(1 + u) where u = 1 + v and v = 9 + x2 so
2
1+ 9+x
dy
dy du dv
1
1
=
=
p 2x.
22v
dx
du dv dx
(1 + u)
so
dy
dx
=
x=4
dy
du
u=6
du
dv
v =25
dv
dx
11
8.
7 10
=
x=4
14. Exercises
p
149. Let y = 1 + x3 and nd dy/dx usin
Continuing this way one nds after n
(22)
u1 un
0
1 applications of the product rule that
= u0 u2 un + u1 u0 u3 un + + u1 u2 u3 u0 .
1
2
n
7.2. The Power rule . If all n factors in the previous paragraph are the same, so that the function f
is the nth powe
6.6. Dierentiating a constant multiple of a function . Note that the rule
(cu)0 = cu0
follows from the Constant Rule and the Product Rule.
by
(21)
6.7. Picture of the Product Rule. If u and v are quantities which depend on x, and if increasing x
x causes
9. Limits and Inequalities
This section has two theorems which let you compare limits of dierent functions. The properties in
these theorems are not formulas that allow you to compute limits like the properties (P1 ). . . (P6 ) from 5.
Instead, they allow
10.5. How to make functions discontinuous. Here is a discontinuous function:
(
x2 if x 6= 3,
f ( x) =
47 if x = 3.
In other words, we take a continuous function like g (x) = x2 , and change its value somewhere, e.g. at x = 3.
Then
lim f (x) = 9 6= 47 = f
Proving lim
x !0
B
sin
=1
The circular wedge OAC contains the
triangle OAC and is contained in the right
triangle OAB .
C
The area of triangle OAC is
sin
tan
1
2
sin .
The area of circular wedge OAC is
The area of right triangle OAB is
1
2
1
.
2
tan .
To see this try to compute the derivative at 0,
| x | | 0|
| x|
f 0 (0) = lim
= lim
= lim sign(x).
x!0 x
x!0 x
x!0
0
We know this limit does not exist (see 7.2)
If you look at the graph of f (x) = |x| then you see what is wrong: the graph has a corner at
There are no innitely small real numbers, and this makes Leibniz notation di cult to justify. In the
20th century mathematicians have managed to create a consistent theory of innitesimals which allows you
to compute with dx and dy as Leibniz and his conte