2.1 Tangents and Derivatives at a Point
Finding a Tangent to the Graph of a Function
To find a tangent to an arbitrary curve y=f(x) at a point P(x0,f(x0)), we
• Calculate the slope of the secant through P and a nearby point
Q(x0+h, f(x0+h)).
• Then investigate the limit of the slope as h0.
Slope of the Curve
If the previous limit exists, we have the following definitions.
Reminder: the equation of the tangent line to the curve at P is
Y=f(x0)+m(x-x0)
(point-slope equation)
Example
(a) Find the slope of the curve y=x2 at the point (2, 4)?
(b) Then find an equation for the line tangent to the curve there.
Solution
Rates of Change: Derivative at a Point
The expression
f ( x  h)  f ( x )
h
is called the difference quotient of f at x0 with increment h.
If the difference quotient has a limit as h approaches zero, that limit is
named below.
Summary
2.2 The Derivative as a Function
We now investigate the derivative as a function derived from f by
Considering the limit at each point x in the domain of f.
If f’ exists at a particular x, we say that f is differentiable (has a
derivative) at x. If f’ exists at every point in the domain of f, we call f
is differentiable.
Alternative Formula for the Derivative
An equivalent definition of the derivative is as follows. (let z = x+h)
Calculating Derivatives from the Definition
The process of calculating a derivative is called differentiation. It can
be denoted by
d
f '( x) or
f ( x)
dx
2
Example. Differentiate f ( x)  x
Example. Differentiate f ( x)  x for x>0.
Notations
There are many ways to denote the derivative of a function y = f(x). Some
common alternative notations for the derivative are
f '( x)  y ' 
dy df
d


f ( x)  D( f )( x)  Dx [ f ( x)]
dx dx dx
To indicate the value of a derivative at a specified number x=a, we use
the notation
f '(a) 
dy
df
d
|x  a 
|x  a 
f ( x) | x  a
dx
dx
dx
Graphing the Derivative
Given a graph y=f(x), we can plot the derivative of y=f(x) by estimating
the slopes on the graph of f. That is, we plot the points (x, f’(x)) in
the xy-plane and connect them with a smooth curve, which
represents y=f’(x).
Example: Graph the derivative of the function y=f(x) in the figure below.
What we can learn from the
graph of y=f’(x)?
Differentiable on an Interval; One-Sided Derivatives
If a function f is differentiable on an open interval (finite or infinite) if it has a
derivative at each point of the interval.
It is differentiable on a closed interval [a, b] if it is differentiable on the
interior (a, b) and if the limits
lim
f ( a  h)  f ( a )
h
Right-hand derivative at a
lim
f (b  h)  f (b)
h
Left-hand derivative at b
h 0
h 0
exist at the endpoints.
A function has a derivative at a point if and only if the left-hand and
right-hand derivatives there, and these one-sided derivatives are equal.
When Does A Function Not Have a Derivative at a Point
A function can fail to have a derivative at a point for several reasons,
such as at points where the graph has
1. a corner, where the one-sided derivatives differ.
2. a cusp, where the slope of PQ approaches  from one side and - 
from the other.
3. a vertical tangent, where the slope of PQ approaches  from both
sides or approaches -  from both sides.
4. a discontinuity.
Differentiable Functions Are Continuous
Note: The converse of Theorem 1 is false. A function need not have
a derivative at a point where it is continuous.
For example, y=|x| is continuous at everywhere but is not differentiable
at x=0.
2.3 Differentiation Rules
The Power Rule is actually valid for all real numbers n.
Examples
Example.
Constant Multiple Rule
d
d
[ x11 ]   [ x11 ]  (11x10 )  11x10
dx
dx
d 
d 2
[ 2 ]   [ x ]   (2 x 3 )  2 x 3
dx x
dx
Note:
d
d
d
du
(u )  (1 u )  1 (u )  
dx
dx
dx
dx
Example.
Derivative Sum Rule
Example.
Derivative Product Rule
In function notation:
d
[ f ( x) g ( x)]  f ( x) g '( x)  g ( x) f '( x)
dx
Example
Example: Find
Solution:
dy
if y  (2 x3  2)(6 x 2  3x).
dx
Derivative Quotient Rule
In function notation:
d f ( x)
g ( x) f '( x)  f ( x) g '( x)
[
]
dx g ( x)
g 2 ( x)
Example
Example:
Solution:
2 x3  x 2  4
Find y '( x) if y 
.
x 5
Second- and Higher-Order Derivatives
The derivative f’ of a function f is itself a function and hence may have a
derivative of its own.
If f’ is differentiable, then its derivative is denoted by f’’. So f’’=(f’)’ and is
called the second derivative of f.
d 2 y d dy
dy '
f ''( x)  2  ( ) 
 y ''  D 2 ( f )( x)  Dx 2 [ f ( x)]
dx
dx dx
dx
Similarly, we have third, fourth, fifth, and even higher derivatives of f.
A general nth order derivative can be denoted by
y
(n)
d ( n 1) d n y

y
 n  Dn y
dx
dx
3
2
Example: If y  4x  x  2x  6, then
2.4 The Derivative as a Rate of Change
Thus, instantaneous rates are limits of average rates.
When we say rate of change, we mean instantaneous rate of change.
Motion Along a Line: Displacement, Velocity, Speed,
Acceleration, and Jerk
Suppose that an object is moving along a s-axis so that we know its
position s on that line as a function of time t: s=f(t).
The displacement of the object over the time interval from t to t+∆t is
∆s = f(t+ ∆t)-f(t);
The average velocity of the object over that time interval is
s f (t  t )  f (t )
vav 

t
t
Velocity
To find the body’s velocity at the exact instant t, we take the limit of the
Average velocity over the interval from t to t+ ∆t as ∆t shrinks to zero.
The limit is the derivative of f with respect to t.
Besides telling how fast an object is moving, its velocity tells the direction of
Motion.
The speedometer always shows speed, which is the absolute value of velocity.
Speed measures the rate of progress regardless of direction
The figure blow shows the velocity v=f’(t) of a particle moving on a
coordinate line., what can you say about the movement ?
Acceleration
The rate at which a body’s velocity changes is the body’s acceleration.
The acceleration measures how quickly the body picks up or loses speed.
A sudden change in acceleration is called a jerk.
Example
Near the surface of the earth all bodies fall with the same constant
acceleration. In fact, we have
s=(1/2)gt2 ,
where s is the distance fallen and g is the acceleration due to Earth’s
gravity.
With t in seconds, the value of g at sea lever is 32 ft/ sec2 or 9.8m/sec2.
Example
Example: Figure left shows the free fall of a
heavy ball bearing released from rest at time
t=0.
(a) How many meters does the ball fall in the
first 2 sec?
(b) What is its velocity, speed, and acceleration
when t=2?
2.5 Derivatives of Trigonometric Functions
Example
Example: Find
Solution:
dy
if y  x cos x.
dx
Example
Example:
Solution:
Find
dy
cos x
if y 
.
dx
1  sin x
Example: A body hanging from a spring is stretched down 5 units beyond
Its rest position and released at time t=0 to bob up and down. Its position
at any later time is s=5cos t. What are its
velocity and acceleration at time t?
Since
tan x 
sin x
cos x
1
1
, cot x 
, sec x 
, csc x 
cos x
sin x
cos x
sin x
We have
Example
Example:
Solution:
Find y '' if f ( x)  tan x.
2.6 Exponential Functions
In general, if a1 is a positive constant, the function f(x)=ax is the
exponential function with base a.
If x=n is a positive integer, then an=a  a  …  a.
If x=0, then a0=1,
1
1 n
n
a


(
)
If x=-n for some positive integer n, then
an
a
1/n
n
If x=1/n for some positive integer n, then a  a
p/q
p
p
If x=p/q is any rational number, then a  a  ( a )
q
If x is an irrational number, then
q
Rules for Exponents
The Natural Exponential Function ex
The most important exponential function used for modeling natural,
physical, and economic phenomena is the natural exponential
function, whose base is a special number e.
The number e is irrational, and its value is 2.718281828 to nine decimal
places.
The graph of y=ex has slope 1 when it crosses the y-axis.
Derivative of the Natural Exponential Function
Example. Find the derivative of y=e-x.
Solution:
Example. Find the derivative of y=e-1/x.
2.7 The Chain Rule
Example
Example: Let y= sin( x 2 ). Find d .
dx
Solution:
“Outside-inside” Rule
It sometimes helps to think about the Chain Rule using functional
notation. If y=f(g(x)), then
dy
 f '( g ( x)) g '( x)
dx
In words, differentiate the “outside” function f and evaluate it at the
“inside” function g(x) left alone; then multiply by the derivative of the
“inside” function.
Example
Example. Differentiate sin(2x+ex) with respect to x.
Solution.
Example. Differentiate e3x with respect to x.
Solution.
In general, we have
For example.
d sin x
d
(e )  esin x
(sin x)  esin x cos x
dx
dx
Example: find derivative of |x| when x ≠ 0.
Repeated Use of the Chain Rule
Sometimes, we have to apply the chain rule more than once to calculate
a derivative.
Example. Find
Solution.
d
[sin(tan 3 x)].
dx
The Chain Rule with Powers of a Function
If f is a differentiable function of u and if u is a differentiable function of x,
then substituting y = f(u) into the Chain Rule formula leads to the
formula
d
du
f (u )  f '(u )
dx
dx
This result is called the generalized derivative formula for f.
For example. If f(u)=un and if u is a differentiable function of x, then we can
Obtain the Power Chain Rule:
d n
du
u  nu n 1
dx
dx
Example
Example:
Solution:
Find
d
( x  2)8
dx
Example
Example:
Solution:
Find
d
[ tan x ].
dx
Example
Example:
Solution:
Find
d
[(1  sec3 x)10 ]
dx
2.8 Implicit Differentiation
Definition. We will say that a given equation in x and y defines the function
f implicitly if the graph of y = f(x) coincides with a portion of the graph of the
equation.
Example:
•The equation
x2  y 2  1 implicitly defines functions
f1 ( x)  1  x 2 and f 2 ( x)   1  x 2
•The equation
f1( x) 
x  y 2 implicitly defines the functions
x and f 2 ( x)   x
Two differentiable methods
There are two methods to differentiate the functions defined implicitly by
the equation.
For example: Find dy / dx if xy  1
1
One way is to rewrite this equation as y  , from which it
x
dy
1
 2
follows that
dx
x
Two differentiable methods
The other method is to differentiate both sides of the equation before
solving for y in terms of x, treating y as a differentiable function of x.
The method is called implicit differentiation.
With this approach we obtain d [ xy ]  d [1]
dx
dx
d
d
x [ y ]  y [ x]  0
dx
dx
dy
x y0
dx
dy
y

dx
x
1
dy
1
Since y  ,
 2
x
dx
x
Implicit Differentiation
Example
Example: Use implicit differentiation to find dy / dx if x2  y 2  3x
Solution:
Example
Example: Find dy / dx if y3  3x 11  0
Solution:
Lenses, tangents and Normal Lines
In the law that describes how light changes direction as it enters a lens,
the important angles are the angles the light makes with the line
perpendicular to the surface of the lens at the point of entry.
surf
This line is called the normal to the
surface at the point of entry.
The normal is the line perpendicular
to the tangent of the profile curve at
the point of entry.
Example
Show that the point (2, 4) lies on the curve x3+y3-9xy=0. Then find the
tangent and normal to the curve there.
Derivatives of Higher Order
Find dy2 /dx2 if 2x3-3y2=8.
2.9 Inverse Functions and Their Derivatives
A function that undoes, or inverts, the effect of a function f is called the
inverse of f.
Examples
Inverse Function
Note the symbol f -1 for the inverse of f is read “f inverse”. The “-1” in f -1 is
not an exponent; f -1 (x) does not mean 1/f(x).
Finding Inverses
The process of passing from f to f -1 can be summarized as a two-step
process.
1. Solve the equation y=f(x) for x. This gives f formula x=f -1(y) where
x is expressed as a function of y.
2. Interchange x and y, obtaining a formula y=f -1(x), where f -1 is
expressed in the conventional format with x as the independent
variable and y as the dependent variable.
Examples
Find the inverse of
(a) y=3x-2.
(b) y=x2,x≥0.
Solution:
Derivative Rule for Inverses
Derivative Rule for Inverses
Example
Let f(x)= x3-2. Find the value of df-1/dx at x=6 = f(2) without finding a formula
for f -1 (x).
2.10 Logarithmic Functions
Natural Logarithm Function
Logarithms with base e and base 10 are so important in applications that
Calculators have special keys for them.
logex is written as lnx
log10x is written as logx
The function y=lnx is called the natural logarithm function, and y=logx is
Often called the common logarithm function.
Properties of Logarithms
Properties of ax and logax
Derivative of the Natural Logarithm Function
Since y=lnx is the inverse function of y=ex, we have
Note:
d
1
[ln x]  , x  0
dx
x
Example: Find
Solution:
d
[ln( x3  4)]
dx
Example
Furthermore, since |x|=x when x>0 and |x|= -x when x<0,
Example: Find d [ln | cos x |]
dx
Solution:
Derivatives of au
Since ax=exlna, we can find the following result.
Note that
Example:
d x
d
d
du
[a ]  a x ln a, [e x ]  e x , [eu ]  eu
dx
dx
dx
dx
Derivatives of logau
Since logax =lnx/lna, we can find the following result.
Note that
Example:
d
1 du
ln u 
dx
u dx
Logarithmic Differentiation
The derivatives of positive functions given by formulas that involve
products, quotients, and powers can often be found more quickly if
we take the natural logarithm of both sides before differentiating.
This process is called logarithmic differentiation.
( x 2  1)( x  3)1/2
Example. Find dy/dx if y 
, x 1
x 1
The Number e as a Limit
2.11 Inverse Trigonometric Functions
The six basic trigonometric functions are not one-to-one (their values
Repeat periodically). However, we can restrict their domains to intervals
on which they are one-to-one.
Six Inverse Trigonometric Functions
Since the restricted functions are now one-to-one, they have inverse,
which we denoted by
y  sin 1 x or y  arcsin x
y  cos 1 x or y  arccos x
y  tan 1 x or y  arctan x
y  cot 1 x or y  arc cot x
y  sec 1 x or y  arc sec x
y  csc 1 x or y  arc csc x
These equations are read “y equals the arcsine of x” or y equals arcsin x”
and so on.
Caution: The -1 in the expressions for the inverse means “inverse.” It does
Not mean reciprocal. The reciprocal of sinx is (sinx)-1=1/sinx=cscx.
Derivative of y = sin-1x
Example: Find dy/dx if
Solution:
y  sin 1 ( x2 )
Derivative of y = tan-1x
Example: Find dy/dx if y  tan 1 (ex )
Solution:
Derivative of y = sec-1x
Example: Find dy/dx if y  sec1 (4 x3 )
Solution:
Derivative of the other Three
There is a much easier way to find the other three inverse trigonometric
Functions-arccosine, arccotantent, and arccosecant, due to the following
Identities:
It follows easily that the derivatives of the inverse cofunctions are the negatives
of the derivatives of the corresponding inverse functions.
2.13 Linearization and Differentials
In general, the tangent to y=f(x) at a point x=a, where f is differentiable,
passes through the point (a, f(a)), so its point-slope equation is
y=f(a)+f’(a)(x-a).
Thus this tangent line is the graph of the linear function L(x)=f(a)+f’(a)(x-a)..
For as long as this line remains close to the graph of f, L(x) gives a good
approximation to f(x).
Linearization
Example
Find the linearization of f(x)=cosx at x=π/2.
Also an important linear approximation for roots and poewrs is
(1+x)k==1+kx (x near 0; any number k).