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Chapter 2
Derivatives
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2.6 Implicit differentiation
1) Explicit function: The function which can be
described by expressing one variable explicitly in terms of
another variable (other variables) are generally called
explicit function---for example, y=xtanx, or
y=[1+x2+x3]1/2 , or in general y=f(x).
2) Implicit function: The functions which are defined
implicitly by a relation between variables--x and y--are
generally called implicit functions--- such as x2+y2 =4, or
7sin(xy)=x2+y3 or, in general F(x,y)=0.If y=f(x) satisfies
F(x, f(x))=0 on an interval I, we say f(x) is a function
defined on I implicitly by F(x,y)=0, or implicit function
defined by F(x,y)=0 .
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3) Derivatives of implicit function Suppose y=f(x)
is an implicit function defined by sin(xy)= x2+y3.
Then sin[xf(x)]=x2+ [f(x)]3. From the equation, we
can find the derivative of f(x) even though we have not
gotten the expression of f(x). Fortunately it is not
necessary to solve the equation for y in terms of x to
find the derivative. We will use the method called
implicit differentiation to find the derivative.
Differentiating both sides of the equation, we obtain
that [f(x)+xf ' (x)]cos[xf(x)]=2x+3[f(x)]2f ' (x). Then
f(x)cos[ xf ( x)]-2 x
y cos(xy)  2 x
f ' ( x) 
 2
2
3[ f ( x)]  xcos[xf ( x)] 3 y  x cos(xy)
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Example (a) If
x3+y3
=27, find
dy
.
dx
(b) Find the equation of the tangent to the
curve x3+y3 =28 at point (1,3).
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Example (a) If x3+y3 =6xy, find y'.
(b) Find the equation of the tangent to the
folium of Descartes x3+y3 =6xy at point (3,3).
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Orthogonal: Two curves are called Orthogonal, if
at each point of intersection their tangent lines are
perpendicular. If two families of curves satisfy that
every curve in one family is orthogonal to every curve in
the another family, then we say the two families of curves
are orthogonal trajectores of each other.
Example The equations xy=c (c0) represents a family
of hyperbolas. And the The equations x2-y2=k (k0)
represents another family of hyperbolas with asymptotes
y=x. Then the two families of curves are trajectores of
each other.
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2.7 Higher derivatives
Derivative f' (x) of differentiable function f(x) is also a
function. If f' (x) is differentiable, then we have [f ' (x)] '.
We will denote it by f ' ' (x), i.e., f' ' (x)=[f ' (x)] '. The new
function f ' ' (x) is called the second derivative of f(x).
If y=f(x), we also can use other notations:
d 2 y d df ( x) d 2 f ( x)
2
2
f ' ' ( x)  y ' '  2  (
)

D
f
(
x
)

D
x f ( x)
2
dx dx
dx
dx
Similarly f ' ' ' (x)=[f '' (x)] '
is called the third derivative
of f(x), and
d 3 y d d 2 f ( x) d 3 f ( x)
3
3
f ' ' ' ( x)  y ' ' '  3  (
)


D
f
(
x
)

D
x f ( x)
2
3
dx dx dx
dx
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And we can define f' ' ' ' (x)=[f ' ' ' (x)] '. From now on
instead of using f' ' ' ' (x) we use f(4)(x) to represent f ' ' ' ' (x).
In general, we define f(n)(x)=[f(n-1)(x)] ', which is called the
nth derivative of f(x). We also like to use the following
notations, if y=f(x),
n
n1
n
d y d d f ( x) d f ( x) n
n
f ( x)  y  n  ( n1 ) 
 D f ( x)  Dx f ( x)
n
dx dx dx
dx
( n)
( n)
Example If y=x4-3x2+6x+9, find y ', y ' ', y ' ' ', y(4).
Example If
1
f(x)= x
, find f(n)(x).
Example If f(x)=sinx, g(x)=cosx, find f(n)(x) and g(n)(x) .
Example Find y ' ' , if x4+y3 =x-y .
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2.8 Related rates (omitted)
2.9 Differentials, Linear and Quadratic
Approximations
Definition: Let x=x-x0, f(x) =f(x)-f(x0). If there
exists a constant A(x0) which is independent of x and
x such that f(x)=A(x0) x+B(x, x0) where
B(x, x 0 )
lim
 0 . Then A x is called
B(x, x0) satisfies x
0
x
differential of f(x) at x0. Generally A x is denoted by
df(x)|x=x = A(x0) x. Replacing x0 by x, the differential
0
is denoted by df(x) and df(x)= A(x) x.
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Corollary: If the differential of f(x) is df(x)= A(x) x,
then f(x) is differentiable and A(x)=f '(x).
Proof: From the definition,
f (t )  f ( x)
A( x)t  B(t , x)
f ' ( x)  lim
 lim
 A( x).
tx
t 0
tx
t
Corollary: (a) If f(x)=x, then dx=df(x)=x.
(b) If f(x) is differentiable, then differential
of f(x) exists and df(x)=f '(x)dx.
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Example (a) Find dy, if y=x3+5x4.
(b) Find the value of dy when x=2 and dx=0.1.
y
Solution:
y=f(x)
Geometric
meaning of
differential of
f(x), df(x)=QS
f(x)=RS
Q
R
dy
P
S
dx=x
o
x
x
t

As x=dx is very small, y=dy ,i.e., f(t)-f(x) f '(x) t.
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Example Use differentials to find an approximate (65)1/3 .
From definition of the differential, we can easily get
If f(x) is differentiable at x=a, and x is very
closed to a, then f(x)  f(a)+f '(a)(x-a). The
approximation is called Linear approximation or
tangent line approximation of f(x) at a. And
function L(x)= f(a)+f '(a)(x-a) is called the
linearization of f(x) at a.
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Example Find the linearization of the function
f(x)=(x+3)1/2 and approximations the numbers (3.98) 1/2
and (4.05)1/2.
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Quadratic approximation to f(x) near x=a:
Suppose f(x) is a function which the second derivative
f ' '(a) exists. P(x)=A+Bx+Cx2 is the parabola which
satisfies P(a)=f(a), P '(a)=f '(a), and P ' '(a)=f ' '(a). As x
is very closed to a, the P(x) is called Quadratic
approximation to f(x) near a.
Corolary: Suppose P(x)=A+Bx+Cx2
is the Quadratic
approximation to f(x) near a. Then
P(x)=f(a)+f '(a)(x-a)+ f' '(a)(x-a)2 / 2.
If P(x) is the quadratic approximation to f(x) near x=a, then as x
is very closed to a, P(x) f(x).That is

f(x) f(a)+f '(a)(x-a)+ f' '(a)(x-a)1/2/2.
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Example Find the quadratic approximation to
f(x)=cosx near 0.
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Example Find the quadratic approximation
to f(x)=(x+3)1/2 near x=1.
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2.10 Newton’s method(to be omitted)
The method is to give a way to get a approximation
to a root of an equation.
Suppose f(x) is defined on [a,b], f '(x) does not
value 0. Let x0[a,b], f(a)f(b)<0.
f(x1 )
And
, x2=x1- f' (x1 ) . Keeping repeating
the process (xn=xn-1- f(x n-1 ) ), we obtain a sequence
f' (x n -1 )
lim xn  r
of approximations x , x ,..., x ,...... If
f(x 0 )
x1=x0- f' (x )
0
1
2
n
n
, then r is the root of the equation f(x)=0.
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Example Starting with x1=2, find the third
approximation x3 to the root of the equation
x3-2x-5=0.
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2.1 Derivatives
We defined the slope of the tangent to a curve with
equation y=f(x) at the point x=a to be
f(a  h) - f(a)
m  lim
h0
h
Generally we give the following definition:
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Definition: The derivative of a function f at
a number a, denoted by f´(a), is
f(a  h) - f(a)
lim
h 0
h
if this limit exists.
Then we have:
f ( x) - f (a )
f ' (a)  lim
x a
x-a
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Example Find the derivative of the function
y=x2-8x+9 at a.
Geometric interpretation: The derivative of
the function y=f(x) at a is the slope of
tangent line to y=f(x) at (a, f(a)). The line
is through (a, f(a)).So if f ´(a) exists, the
equation of the tangent line to the curve
y=f(x) at (a, f(a)) is y-f(a)= f ´(a) (x-a).
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en
Example Find the equation of the tangent line
of the function y=x2-3x+5 at x=1.
In the definition if we replace a by x, then we
obtain a new function f ´(x) which is deduced
from f(x).
f ( x  h)-f ( x)
f ' ( x)  lim
h 0
h
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Example If f(x)=(x-1)1/2, find the derivative of
f . State the domain of f´(x).
Example Find the derivative of f if
1-x
f(x)= 2+x
Other notations: If y=f(x), then the other notations
d
dy
df
are that f´(x)= y´=dx =dx dx
= f (x) =Df(x)=Dxf(x).
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The symbol D and d/dx are called differential operators.
We also use the notations:
df
df 
f ' ( a) 
 
dx xa dx  xa
Definition A function f is called differentiable at
a if f´(a) exists. It is differentiable on an open
interval (a,b) [or (a,+) or (- ,b) ] if it is
differentiable at every number in the interval.
Example Where is the function f(x)=|x| is
differentiable?
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Theorem: If f(x) is differentiable at a, then f(x) is
continuous at a.( The converse is false)
There are several cases a function fails to be differentiable
(1) the points at which graph of the function f has “corners”,
such as f(x)=|x| at x=0;
(2) the points at which the function is not continuous, such as, the
function, defined as f(x)=2x for x1, and 3x for x<1, at x=1;
(3) the points at which the curve has a vertical tangent line,
such as, f(x)=x1/3, at x=0.
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2.2 Differentiation
1). Theorem If f is a constant function, f(x)=c,
d
then f´(x)= (c)´=0, i.e., dx c =0.
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2). The power rule If f(x)=xn, where n is a
d n
n-1
positive integer, then f´(x)= nx , dx x =nxn-1.
Example If f(x)=x100, find f´(x).
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3)Theorem Suppose c is a constant and f´(x) and
g´(x) exist.Then
(a) (cf(x))´ exists and (cf(x))´=cf´(x);
(b) (f(x)+g(x))´exists and (f(x)+g(x)´=f´(x)+g´(x);
(c) (f(x)-g(x))´exists and (f(x)-g(x)´=f´(x)-g´(x).
Example If f(x)= x50 +x100, find f´(x).
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4) Product rule Suppose f´(x) and g´(x) exist.
Then
f(x)g(x) is differentiable and
[f(x)g(x)]´= f´(x) g(x)+f(x)g´(x) .
Example If f(x)= (2x5)(3x10), find f´(x).
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4) Quotient rule Suppose f´(x) and g´(x) exist and
g(x)0, then f(x)/g(x) is differentiable and
[f(x)/g(x)]´= [f´(x) g(x)-f(x)g´(x)]/[g(x)]2.
x2+2x-5
Example If f(x)= x3-6 , find f´(x).
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2). The power rule (general version) If f(x)=xn,
where n is any real number, then f´(x)= nxn-1,
,i.e.,
d
dx
xn =nxn-1.
Example If f(x)=x, find f´(x).
If g(x)= x1/2, g´(x)=?
Example Differentiate the function f(t)=(1-t)t1/3.
Table of differentiation formulas (in paper 119)
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2.3 Rate of change in the Economics
Suppose C(x) is the total cost that a company
incurs in producing x units of certain commodity.
The function C is called a cost function. If the
number of items produced increased from x1 to x2,
the additional cost is C= C(x2)-C(x1), and the
average of change of the cost is
C = C(x2)-C(x1) = C(x1+x)-C(x1)
x2-x1
x
x
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The limit of this quantity as x0, is
called the marginal cost by economist.
C dC

Marginal cost= lim
x    x
dx
Taking x=1 and n large (so that x is small
compared to n),we have C'(n) C(n+1)-C(n).

Thus the marginal cost of producing n is
approximately equal to the cost of
producing one more unit [the (n+1)st unit].
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2.4 Derivatives of trigonometric functions
sin   0
(1) Theorem lim
 0
Proof: suppose OP=1
and (0, /2). So we will
show
y
B D
lim sin   0.
 0
A


o
C
x
Notice that 0<|BC|<arcAB
cos   1.
(2) Corollary lim
 0
-1
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(3) Theorem lim sin   1.
 0

Proof: Notice that
Area of OAB<Area of
sector OAB<Area of OAD.
y
B D

o
(4) Corollary lim cos   1  0.
 0

A
C x
-1
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sin 7 x
.
Example Find lim
x0 sin 4 x
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(5)
Theorem
Example
(6)
Differentiate y=xsinx.
Theorem
Example
d
sin x  cos x.
dx
d
cos x   sin x.
dx
Differentiate y=tanx.
Corollary (tanx)'=sec2x
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Example
Differentiate y=cotx.
Corollary (cotx)'= - csc2x
tan x  sin x  cot x
Example Differentiate f(x)= 2  x2  cos x .
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2.5 Chain rule
The chain rule If the derivative g'(x) and
derivative f '(u), with respect to u, exist,
then the composite function f(g(x)) is
differentiable, and [f(g(x))] ' =f '(g(x))g '(x).
Proof: Let u=g(x+x)-g(x)
y=f(u+u)-f(u)
dy
y
y u
 lim
 lim
.
dx x0 x x0 u x
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Case 1: du/dx0, then u0
dy
y
y
u
 lim
 lim
 lim
.
dx x0 x x0 u x0 x
Case 2: du/dx=0. there are two cases:
dy
y
y
u
 lim
 lim
 0.
(a) u 0, dx  lim
x 0 x
x 0 u x 0 x
dy
y
0
 lim
 0.
(b) u= 0, dx  lim
x 0 x
x 0 x
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Example
Find F '(x) if F(x)=(1+x2)3/4.
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