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CO4Discuss and apply comprehensively the concepts, properties and theorems of functions, limits, continuity and the derivatives in determining the derivatives of algebraic functions
Objective:
At the end of the discussion, the students should be able to evaluate limits and determine the derivative of a continuous algebraic function given in the explicit or implicit form.
The Calculus
Calculus- Is the mathematics of change- two basic branches: differential and integral
calculus
Lesson 1 : Functions and Limits
FUNCTIONS
- A relation between variables x and y is a rule of correspondence that assigns an element x from the Set A to an element y of Set B.
- A function f from set A to set B is a rule of correspondence that assigns to each element x in the set A exactly one element y in the set B. It is a set of ordered pairs ( x, y) such that no two pairs will have the same first element.
Domain (set of all x’s) Range (set of all y’s)
1
2
3
4
5
2
10
8
6
4
Mapping of X- values into y-values ( 1 -1 correspendence)
X Y
X = Set A
Domain Set
1
2
3
4
5
Y = Set B
Range Set
2
1086
4
Mapping illustrating many – 1 correspondence
1
2
3
4
5
2
1086
4
Mapping of the elements of Set A in to Set B illustrating 1 – many type of correspondence.
X = Set A Y = Set B
Function Notation
We commonly name a function by letter with f the most commonly used letter to refer to functions. However, a function can be referred to by any letter.
The function called f
The independent variable, x
f(x) defines a rule express in terms of x as given by the right
hand side expression.
Note: The value of the function f(x) is determined by substituting x- value into the expression.
PIECEWISE DEFINED FUNCTION
A piecewise defined function is function defined by different formulas on different parts of its domain; as in,
Graph of a Function
The graph of a function f consists of all points (x, y) whose coordinates satisfy y = f(x), for all x in the domain of f. The set of ordered pairs (x, y) may also be represented by (x, f(x)) since y = f(x).
Recall: The Vertical Line Test
A set of points in a coordinate plane is the graph of a function y = f(x) if and only if no vertical line intersects the graph at more than one point.
ODD and EVEN FUNCTIONS
A function is an even function if and only if The graph of an even function is symmetric with respect to the y-axis.
A function is an odd function if and only if The graph of an odd function is symmetric with respect to the origin.
Sample Problems
For each of the following, determine the domain and range, then sketch the graph.
1x if 1x21x if x
)x(f d.
3xx2)x(f g. x)x(f .c
1x3x)x(f .f x1)x(f .b
4t4t4
4t
ififif
t1t
3)t(f e. 5x3)x(f .a
2
2
with domain the set of all x in the domain of g such that g(x) is in the domain of f or in other words, whenever both g(x) and f(g(x)) are defined.
In the same way,
with domain as the set of all x in the domain of f such that f(x) is in the domain of g, or, in other words, whenever both f(x) and g(f(x)) are defined.
The composition function, denoted by , is defined as
For each of the following pair of functions:
a) f(x) = 2x – 5 and g(x) = x2 – 1
b) and
determine the following functions: a) f + g b) f - g c) fg d) f/g e) g/f
f) g) domain of each resulting functions.
Sample Problems
Limits
Informal Definition: If the values of f(x) can be made as close as possible to some value L by taking the value of x as close as possible, but not equal to, a, then we write
Read as “ the limit of f(x) as x approaches a is L” or “ f(x) approaches L as x approaches a”. This can also be written as
Formal Definition of a Limit of a Function:
Let f be a function defined at every number in some open interval containing a , except possibly at the number a itself. The limit of f(x) as x approaches a is L , written as,
If given any
Geometrically, this can be viewed as follows:
Theorem 1: Limit of a Constant If c is a constant, then for any number a
Theorem 2: Limit of the Identify Function Theorem 3: Limit of a Linear Function
If m and b are constants
Theorem 4: Limit of the Sum or Difference of Functions
Theorems on Limits
Theorem 5: Limit of the Product
Theorem 6: Limit of the nth Power of a function
Theorem 7: Limit of a Quotient
Theorem 8: Limit of the nth Root of a Function
Using the theorems on Limits, evaluate each of the following:
1. 6. Let
2. find:
3.
4.
5.
Sample Problems
)t(glim ),t(glim2t0t
Definition of One-Sided LimitsInformal Definition:If the value of f(x) can be made as close to L by taking the value of x sufficiently close to a , but always greater than a , then
read as “the limit of f(x) as x approaches a from the right is L.”Similarly, if the value of f(x) can be made as close to L by taking the value of x sufficiently close to a , but always less than a , then
read as “the limit of f(x) as x approaches a from the left is L.”If both statements are true and equal then .
Geometrically,
Infinite Limits
The expressions
denote that the function increases/decreases without bound as x approaches a from the right/ left and that f(x) has infinite limit.
A function having infinite limit at a exhibits a vertical asymptote at x = a.
x=a
0
x=a
0
Geometrically;
Limits at Infinity
If as x increases/decreases without bound, the value of the function f(x) gets closer and closer to L then
If L is finite, then limits at infinity is associated with the existence of a horizontal asymptote at y = L.
Geometrically,
Y=L
0
Y=L0
LIMITS OF FUNCTIONS USING THE SQUEEZE PRINCIPLE The Squeeze Principle is used on limit problems where the usual algebraic methods (factoring, conjugation, algebraic manipulation, etc.) are not effective. However, it requires that you be able to ``squeeze'' your function in between two other ``simpler'' functions whose limits are easily evaluated and equal. The use of the Squeeze Principle requires accurate analysis, deft algebra skills, and careful use of inequalities.
Theorem:
Sample Problems
3xxcos2lim .2
x
2xxlim .9
x)xcos1(xsinlim .8
x31
x31
lim .7
4x35xlim .6
2x
20x
0x
4x
Sample Problems
x
- x
23
x
3
23
x
0x
e52lim .141x4
x5lim .13
2x4x9x4x2x7lim .12
x2cos1x4cos1lim .11
Lesson 2 : Continuity of Function
If one or more of the above conditions fails to hold at c the function is said to be discontinuous at c. A function that is continuous on the entire real line is said to be continuous everywhere.
DEFINITION: CONTINUITY OF A FUNCTION
Definition:A function f(x) is said to be conrinuous at x = c if
and only if the following conditions hold:
If functions f and g are continuous functions at x = c, then the following are true:
a. f + g is continuous at cb. f – g is continuous at cc. fg is continuous at cd. f/g is continuous at c provided g( c ) is not zero.
The figure above illustrates that the function is discontinuous at x=c and violates the first condition.
The figure above illustrates that the function is discontinuous at x = c and violates the second condition. This kind of discontinuity is called jump discontinuity.
Types of Discontinuity
The figure above illustrates that the limit coming from the right and left of c are both undefined, thus the function is discontinuous at x = c and violates the second condition. This kind of discontinuity is called infinite discontinuity.
The figure above shows that the function is defined at c and that the limit coming from the right and left of c both exist thus the two sided limit exist. However,Thus, the function is discontinuous at x = c, violating the third condition. This kind of discontinuity is calledremovable discontinuity ( missing point).
1. Investigate the discontinuity of the function f defined. What type of discontinuity is illustrated?
a)
b)
Show the point(s) of discontinuity by sketching the graph of the function .
Sample Problems
2. Find values of the constants k and m, if possible, that will make the function f(x) defined as
be continuous everywhere.
1x2x1
2x
7xx2k)1x(m
5x)x(f
3
2
Lesson 3: The Derivative
Derivative of a Function
The process of finding the derivative of a function is called differentiation and the branch of calculus that deals with this process is called differential calculus. Differentiation is an important mathematical tool in physics, mechanics, economics and many other disciplines that involve change and motion.
y
))(,( 11 xfxP ))(,( 22 xfxQ
)(xfy
xxxxxx
12
12
tangent line
secant line
x
y
Consider:-Two distinct points P and Q-Determine slope of the secant line PQ- Investigate how the slope changes as Q approaches P.- Determine the limit of the secant line as Q approaches P.
DEFINITION:Suppose that is in the domain of the function f, the tangent line to the curve at the point is with equation
1x)(xfy ))(,( 11 xfxP
)()( 11 xxmxfy
where provided the limit exists, and
is the point of tangency.
))(,( 11 xfxP
xxfxxfm
x
)()(lim 110
DEFINITION
The derivative of at point P on the curve is equal to the slope of the tangent line at P, thus the derivative of the function f with respect to x, given by , at any x in its domain is defined as:
)(xfy
0 0
( ) ( )lim limx x
dy y f x x f xdx x x
provided the limit exists.
)(' xfdxdy
Note: A function is said to be differentiable at if the derivative of y wrt x is defined at .
0x0x
Other notations for the derivative of a function:
)(),(',','),(, xfdxdandxffyxfDyD xx
Note: To find the slope of the tangent line to the curve at point P means that we are to find the value of the derivative at that point P.
THE Derivative of a Function based on the Definition ( The four-step or increment method)
To determine the derivative of a function based on the definition (increment method or more commonly known as the four-step rule) , the procedure is as follows:
xx
yy STEP 1: Substitute for x and for y in )(xfy
STEP 2: Subtract y = f(x) from the result of step 1 to obtain in terms of x and y .x
STEP 3: Divide both sides of step 2 by .x
STEP 4: Find the limit of the expression resulting from step 3 as approaches 0.
x
Sample Problems
Find the derivative of each of the following functions based on the definition:
DERIVATIVE USING FORMULA
Finding the derivative of a function using the definition or the increment-method (four-step rule) can be laborious and tedious specially when the functions to be differentiated are complex. The theorems on differentiation will enable us to calculate derivatives more efficiently and hopefully will make calculus easy and enjoyable.
DIFFERENTIATION FORMULA
1. Derivative of a ConstantTheorem: The derivative of a constant function
is 0; that is, if c is any real number, then .
0][ cdxd
2. Derivative of a Constant Times a FunctionTheorem: ( Constant Multiple Rule) If f is a differentiable function at x and c is any real number, then is also differentiable at x and
)()( xf
dxdcxcf
dxd
cf
3. Derivatives of Power FunctionsTheorem: ( Power Rule) If n is a positive integer, then .
1][ nn nxxdxd
DIFFERENTIATION FORMULA
4. Derivatives of Sums or Differences Theorem: ( Sum or Difference Rule) If f and g are both differentiable functions at x, then so are f+g and f-g , and or
gdxdf
dxdgf
dxd
)()()()( xg
dxdxf
dxdxgxf
dxd
5. Derivative of a ProductTheorem: (The Product Rule) If f and g are both differentiable functions at x, then so is the product , and
or
gf
dxdfg
dxdgfgf
dxd
)()()]([)()()( xfdxdxgxg
dxdxfxgxf
dxd
DIFFERENTIATION FORMULA6. Derivative of a QuotientTheorem: (The Quotient Rule) If f and g are both differentiable functions at x, and if then is differentiable at x and
or 2gdxdgf
dxdfg
gf
dxd
0ggf
2)(
)()()()(
)()(
xg
xgdxdxfxf
dxdxg
xgxf
dxd
7. Derivatives of Composition ( Chain Rule)Theorem: (The Chain Rule) If g is differentiable at x and if f is differentiable at g(x) , then the composition is differentiable at x. Moreover, if y=f(g(x)) and u = g(x) then y = f(u) and
gf
DIFFERENTIATION FORMULA
9. Derivative of a Radical with index equal to 2
If u is a differentiable function of x, then u
dxdu
udxd
2
10. Derivative of a Radical with index other than 2If n is any positive integer and u is a differentiable function of x, then dx
duun
udxd nn
111 1
8. Derivative of a PowerIf u is a differentiable function of x and n is any real number , then
Implicit Differentiation
On occasions that a function F(x , y) = 0 can not be defined in the explicit form y = f(x) then the implicit form F ( x , y) = 0 can be used as basis in defining the derivative of y ( the dependent variable) with respect to x ( the independent variable).
When differentiating F( x, y) = 0, consider that y is defined implicitly in terms of x , then apply the chain rule. As a rule,1. Differentiate both sides of the equation with respect to x.2. Collect all terms involving dy/dx on the left side of the
equation and the rest of the terms on the other side.3. Factor dy/dx out of the left member of the equation and solve
for dy/dx by dividing the equation by the coefficient of dy/dx.
Higher Order DerivativeThe notation dy/dx represent the first derivative of y with respect to x. And if dy/dx is differentiable, then the derivative of dy/dx with respect to x gives the second order derivative of y with respect to x and is denoted by .
Given:
Sample Problems
Differentiate y with respect to x. Express dy/dx in simplest form.
Sample Problems
Determine the derivative required:
DIFFERENTIATION OF LOGARITHMIC
FUNCTIONS
TRANSCENDENTAL FUNCTIONS
Kinds of transcendental functions:1. logarithmic and exponential functions2. trigonometric and inverse trigonometric
functions3. hyperbolic and inverse hyperbolic functions
Note:Each pair of functions above is an inverse to each other.
A logarithmic function with the base a, a>0 and a1 is defined by
The LOGARITHMIC FUNCTIONS
ya ax ifonly and if xlogy
y
a
a x
logy form
formlExponentia
xcLogarithmi
Logarithmic Form Exponential Form
214log16 2
1
164
38log2 328
EXAMPLE:
SOME LOGARITHMS OF KNOWN BASES:
NATURAL LOGARITHMSLogarithms to the base e = 2.718 are called natural logarithms (from the Latin word Logarithmic Naturalis or Napieran logarithms).
COMMON LOGARITHMSLogarithms to the base 10 are called common logarithms.
xlnxlog e
xlogxlog 10
CHANGE OF BASEWhen the base of a logarithm is other than e or 10, express its equivalent using the base e or 10 in the formula
blnxln
blogxlogxlog or
blogxlogxlog
e
eb
a
ab
3 lny ln
3logylogylog or
3logylogylog .2
2 lnx ln
2logxlogxlog or
2logxlogxlog .1
:Example
e
e3
10
103
e
e2
10
102
yx then ,ylogxlog If .8 palog .7 1alog .6
01log .5 Nlogp1NlogNlog .4
NlogpNlog .3
logarithms 2 of quotient a not NlogMlog
NMlog:Note
NlogMlogNMlog .2
vedistributi not NlogMlogNMlog :Note NlogMlogMNlog 1.
1a and p,,N,M numbers positive For
aa
paa
aap1
ap
a
ap
a
a
aa
aaa
aaa
aaa
DIFFERENTIATION FORMULA
Derivative of Logarithmic Function The derivative of the logarithmic function for any given base and any differentiable function of u
f(x)u where; dxdu
u1)u (ln
dxd
u lnulog and 1elog but dxdu elog
u1)u(log
dxd
f(x)u where; dxdu elog
u1)u(log
dxd
eeee
aa
:e base For
:abase given any For
1x4lny .1
3x21ln)x(f .2
x531x3lnxf .3 2
x3
xlnxh .44
x4xlnlnxG .5
2
3
2
xln1x4lnxg .6
xlnxlny .7 2
22 x1x1xlnxxF .8
A. Find the derivative of each of the following natural logarithmic functions and simplify the result:
Answers on A :
1x44'y .1
12x
6xf' or x21
6x'f .2
x531x35x9x30x'f .3 2
2
x3x4
x13x'h .4
xlnxxln1x'G .5
xln1x4x31x4ln1x4xlnx8x'g .6 22
222
xlnxx1x2'y .7 2
2
2x1x lnx'F .8
x3logy .1 2 2
22 xlog3logxh .2
22 y3logyf .3
2x x3logxg .4
3 23 4xlogxF .5
1zz34logzH .6
2
5
1t3
3t23tlogtG .7 2
2
B. Differentiate the following logarithmic functions.
yx5 xy ln .1
1yx lnyx ln .2
C. Find the derivative using implicit differentiation.
Answers on B :elog
x1'y .1 2
elog2x1x'h .2 2
elog2y1y'f .3 2
ex
x xlog21'g .4
elog
32
4xxx'F .5 32
elog21
4z4z3z34z6z3z'H .6 523
2
elog31t33t23t
3t20t3t2t'G .7 2
23
Answers on C : y1x
1xy'y .1
xy'y . 2
Logarithmic Differentiation
Oftentimes, the derivatives of algebraic functions which appear complicated in form (involving products, quotients and powers) can be found quickly by taking the natural logarithms of both sides and applying the properties of logarithms before differentiation. This method is called logarithmic differentiation.
1. Take the natural logarithm of both sides and apply the properties of logarithms.
2. Differentiate both sides and reduce the right side to a single fraction.
3. Solve for y’ by multiplying the right side by y.4. Substitute and simplify the result.
Steps in applying logarithmic differentiation.
NOTE: Logarithmic differentiation is also applicable whenever the base and its power are both functions.
2x5
1x3xy if dx
dy Find .121
2
2x51x3x236x23x15
y'y
2
3
EXAMPLE:
ationdifferenti clogarithmi using
2x5ln1xln213xln
2x51x3xlnyln
:Solution
221
2
52x5
111x
121x2
3x1
y'y
2
2x51x3x2
1x3x252x53x2x51x2x2y'y
2
22
2x51x3x230x30x10x106x15x2x5x8x20x8x20
y'y
2
2323223
y2x51x3x2
36x23x15'y 2
3
221
3
2x51x2
36x23x15'y
2x51x3x
2x51x3x236x23x15'y
21
2
2
3
5 32 3x41xy if dx
dy Find .2
1x41x
x6x12x2051
1x41xx12x12x6x8
51
y'y
32
24
32
244
EXAMPLE:
ationdifferenti clogarithmi using
3x41xln513x41xlnyln
:Solution
3251
32
3x4ln1xln51yln 32
232 x12
3x41x2
1x1
51
y'y
3x41x1xx123x4x2
51
y'y
32
223
y
xxxxx'y
34136102
51
32
3
5132
32
3341
34136102
51
xxxx
xxx'y
151
323 3x41x3x6x10x251'y
54
323 3x41x3x6x10x52'y
xxy if dx
dy Find .1
xlnxylnxlnyln x
Logarithmic differentiation is also applicable whenever the base and its power are both functions. (Variable to variable power.)
Example:
1xln1x1x'y
y1
xx y butyxln1'y
xxxln1'y
1x2ln1xyln
1x2lnyln 1x
1x1x2y if dx
dy Find .2
11x2ln21x2
11x'yy1
1x2ln1x21x2'y
y1
1-x12x y buty1x2ln1x21x2'y
1-x12x 1x2
1x2ln1x21x2'y
1-1-x12x 1x2ln1x21x2'y
2-x12x 1x2ln1x21x2'y
x5x6y .3
5x6lnxyln
5x6lny lnx
x2
15x6ln5x62
65x6
1xy'y1
12x
5x6x2
5x6ln5x6x6y'
x25x6ln
5x6x3y'
y1
5x6x2
5x6ln5x6x6y'y1
x5x6y buty
5x6x25x6ln5x6x6y'
x5x6
5x6x25x6ln5x6x6y'
1xx34y .4
x34ln1xyln
x34lny ln 1x
1x21x34ln3
x3411x'y
y1
1x2x34ln
x341x3'y
y1
1xx342
x34lnx341x6'yy1
1xx34y but y1xx342
x34lnx341x6y'
1xx341xx342
x34lnx341x6'y
11xx341x2
x34lnx341x6y'
2xlog3x4xh .1 22
3233 x3xlogxf .2
234 x11x3logxg .3
3 24 2x3xxlnxF .4
)x43log(
x4x3xH .5 3
2
36
4 1xlogx3y .6
A. Differentiate and simplify each of the following:EXERCISES:
B. Differentiate each of the following using implicit differentiation.
1x2
3x4xx3y .1 2
32
5 32 3x41xxf .2
32 x4logyxxylog .1
x2elnylnxxlny .2
C. Differentiate each of the following using logarithmic differentiation.
DIFFERENTIATION OF EXPONENTIAL FUNCTIONS
The EXPONENTIAL FUNCTION
.ylogx as writtenbe also may ay function, c logarithmi
of inverse the is function l exponentia the Sincenumber. real a is x whereayby defined is 1,a
and 0a a, base withfunction l exponentia The
a
x
x
.
.
nmanama .1
n m if ,m-na1
n m if , 1n m if ,nma
nama .2
mnanma .3
nbnanab .4
nbnan
ba .5
0 a provided ,10a .6 n1ma
mn1anma .7
Laws of Exponents
xa .8 xloga
y x then aa if .9 yx
DIFFERENTIATION FORMULA
Derivative of Exponential Function The derivative of the exponential function for any given base and any differentiable function of u.
f(x)u where;dxdue )e(
dxd
f(x)u where;dxdualna )a(
dxd
uu
uu
:e base For
:abase given any For
A. Find the derivative of each of the following and simplify the result:
2x3exf .1
x21exg .2
x/12ex4xh .3 2
2
x3
x3
xe6x'f
x6ex'f
x212
2ex'g x21
x2ex
1ex4x'h x/12
x/12
EXAMPLE:
x21x21
x21ex'g
x21
x21e4x'h x/1
1x2e4x'h x/1
x21
x21ex'gx21
2yx2xxye .4
02y'yx1yx21y'xyxye
2y'xyyx2yxye'yxyxe
'xyy2xy2xye3y'yxye2xy
2xy2xye3yyxxye2xy'y
xye2y1x
xye2yxy21y'y
5x42x37y .5
5x42x3dx
d7ln5x42x37'y
4x67ln5x42x37'y
5x42x377ln2x32'y
2x34lnxh .6
2x34
2x34dxd
)x('h
2x34
2x3dxd4ln2x34
x'h
x64lnx'h 4lnx6x'h
2x34lnxh
4ln2x3xh
2xdx
d4ln3x'h
x24ln3x'h
4lnx6x'h
OR
3x2e1xelogxG .6
3elog1elogxG x2x
elog3e2eelog
1eex'G x2
x2
x
x
elog
3e1e1ee23eex'G x2x
xx2x2x
eloge3e1e
e2e23ex'G xx2x
xx2x2
eloge3e1e3e2e3x'G x
x2x
xx2
24 xx3 52xf .7
4224 x3xxx3 2dxd55
dxd2x'f
3x3xxx3 x122ln25x25ln52x'f4224
2lnx65ln52x2x'f 2xx3 24
x2lnx65ln52x'f 2x1x3 24
24 xx3 52xf
24 xx3 52lnxfln
24 xx3 5ln2lnxfln
5lnx2lnx3xfln 24
x25lnx42ln3xfx'f 3
5ln2lnx6x2xfx'f 2
5ln2lnx6x252)x('f 2xx3 24
x5ln2lnx652x'f 2x1x3 24
OR
yx53 .8 4yx
'yx4'y5ln53ln3 3yx
3ln3x415ln5'y x3y
15ln5
3ln3x4'y y
x3
A. Find the derivative and simplify the result.
1x3x2
3xg .1
22 xlnxexf .2
2eey .3 x3
x4
3x222 3xlogxh .4
2x5xG.1
2lnyxxeye.2 22yx
2X1xxH.3
1x2ey.4
1x2
x2x2 eelnxf.5
B. Apply the appropriate formulas to obtain the derivative of the given function and simplify.
EXERCISES:
Logarithmic DifferentiationOftentimes, the derivatives of algebraic functions which appear complicated in form (involving products, quotients and powers) can be found quickly by taking the natural logarithms of both sides and applying the properties of logarithms before differentiation. This method is called logarithmic differentiation.
1. Take the natural logarithm of both sides and apply the properties of logarithms.
2. Differentiate both sides and reduce the right side to a single fraction.
3. Solve for y’ by multiplying the right side by y.4. Substitute and simplify the result.
Steps in applying logarithmic differentiation.
Logarithmic differentiation is also applicable wheneverthe base and its power are both functions.
xxy if dx
dy Find .1
xlnxylnxlnyln x
Logarithmic differentiation is also applicable whenever the base and its power are both functions. (Variable to variable power.)
Example:
1xln1x1x'y
y1
xx y butyxln1'y
xxxln1'y
1x2ln1xyln
1x2lnyln 1x
1x1x2y if dx
dy Find .2
11x2ln21x2
11x'yy1
1x2ln1x21x2'y
y1
1-x12x y buty1x2ln1x21x2'y
1-x12x 1x2
1x2ln1x21x2'y
1-1-x12x 1x2ln1x21x2'y
2-x12x 1x2ln1x21x2'y
x5x6y .3
5x6lnxyln
5x6lny lnx
x2
15x6ln5x62
65x6
1xy'y1
12x
5x6x2
5x6ln5x6x6y'
x25x6ln
5x6x3y'
y1
5x6x2
5x6ln5x6x6y'y1
x5x6y buty
5x6x25x6ln5x6x6y'
x5x6
5x6x25x6ln5x6x6y'
1xx34y .4
x34ln1xyln
x34lny ln 1x
1x21x34ln3
x3411x'y
y1
1x2x34ln
x341x3'y
y1
1xx342
x34lnx341x6'yy1
1xx34y but y1xx342
x34lnx341x6y'
1xx341xx342
x34lnx341x6'y
11xx341x2
x34lnx341x6y'
References
Calculus, Early Transcendental Functions, by Larson and EdwardsCalculus, Early Transcendentals, by Anton, Bivens and DavisUniversity Calculus, Early Transcendentals 2nd ed, by Hass, Weir and ThomasDifferential and Integral Calculus by Love and Rainville
