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Tangent lines Recall: tangent line is the limit of secant line The tangent line to the curve y=f(x) at the point P(a,f(a))
is the line through P with slope
provided that the limit exists. Remark. If the limit does not exist, then the curve does
not have a tangent line at P(a,f(a)).
h
afhaf
ax
afxfm
hax
)()(lim
)()(lim
0
Tangent lines Ex. Find an equation of the tangent line to the hyperbola
y=3/x at the point (3,1).
Sol. Since the limit
an equation of the tangent line is
or simplifies to
)3(3
11 xy
0
( ) ( ) ( ) ( ) 1lim lim
3x a h
f x f a f a h f am
x a h
.063 yx
Velocities Recall: instantaneous velocity is limit of average velocity Suppose the displacement of a motion is given by the
function f(t), then the instantaneous velocity of the motion at time t=a is
Ex. The displacement of free fall motion is given by
find the velocity at t=5. Sol. The velocity is
h
afhafv
h
)()(lim
0
,5.0)( 2gtts
49555.0)5(5.0
lim)5()5(
lim22
00
g
h
ghg
h
shsv
hh
Rates of change Let The difference
quotient
is called the average rate of change of y with respect to x. Instantaneous rate of change =
Ex. The dependence of temperature T with time t is given by the function T(t)=t3-t+1. What is the rate of change of temperature with respective to time at t=2?
Sol. The rate of change is
).()(, 000 xfxxfyxxx
x
xfxxf
x
y
)()( 00
x
yx
0
lim
0 0
(2 ) (2)lim lim 11.t t
T T t T
t t
Definition of derivative Definition The derivative of a function f at a number a,
denoted by is
if the limit exists.
Similarly, we can define left-hand derivative and right-
hand derivative
exists if and only if both and exist and
they are the same.
),(af
h
afhafaf
h
)()(lim)(
0
)(af ).(af
)(af )(af )(af
Example Ex. Find the derivative given
Sol. Since does not exist,
the derivative does not exist.
),0(f
.00
01
cos)(x
xx
xxf
hh
fhfhh
1coslim
)0()0(lim
00
)0(f
Example Ex. Determine the existence of of f(x)=|x|.
Sol. Since
does not exist.
)0(f
).0(1lim)0()0(
lim)0(
,1lim)0()0(
lim)0(
00
00
fh
h
h
fhff
h
h
h
fhff
hh
hh
)0(f
Continuity and derivative Theorem If exists, then f(x) is continuous at x0.
Proof.
Remark. The continuity does not imply the existence of derivative.
For example,
)( 0xf
0limlim)(limlim0000
x
fx
x
fxf
xxxx
.00
01
cos)(x
xx
xxf
Interpretation of derivative The slope of the tangent line to y=f(x) at P(a,f(a)), is the
derivative of f(x) at a,
The derivative is the rate of change of y=f(x) with respect to x at x=a. It measures how fast y is changing with x at a.
).(af
)(af
Derivative as a function Recall that the derivative of a function f at a number a is
given by the limit:
Let the above number a vary in the domain. Replacing a by variable x, the above definition becomes
If for any number x in the domain of f, the derivative
exists, we can regard as a function which assigns to x.
h
afhafaf
h
)()(lim)(
0
)(xf h
xfhxfxf
h
)()(lim)(
0
)(xf f
Remark Some other limit forms
0 0
( ) ( )( ) lim lim
h x
f x h f x yf x
h x
ExampleFind the derivative function of
Sol. Let a be any number, by definition,
Letting a vary, we get the derivative function
.)( nxxf
.)(lim
lim)()(
lim)(
11221
nnnnn
ax
nn
axax
naaxaaxx
ax
ax
ax
afxfaf
.)( 1 nnxxf
Other notations for derivative If we use y=f(x) for the function f, then the following notat
ions can be used for the derivative:
D and d/dx are called differentiation operators.
A function f is called differentiable at a if exists. f is differentiable on [a,b] means f is differentiable in (a,b) and both and exist.
)()()()( xfDxDfxfdx
d
dx
df
dx
dyyxf x
)(af
)(af )(bf
axax dx
dy
dx
dyaf
)(
