Calculus 12
LG 1 – 3
Name
_____________
Worksheet Package
Part A:
1. Find the first derivative of each function:
a)!! y = 3x 2 ! 5!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!b)!! y = 8x ! 2
c)!! f (x) = 6x 2 ! 3x + 2!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!d)!! y = !x 2 + 6
e)!!g(x) = ! x 3 + 6x ! 3!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! f )!!h(x) = 5! 2 x 4 + 6x 2 ! 3! 4
g)!!k(x) =
1 8 2 6 2 4 3
x ! x + x ! !!!!!!!!!!!!!!!!!h)!! y = 6! 3 ! 8! 2 + 24
4
3
5
4
2. Given y, find
dy
:
dx
1
2
1
a)!! y = 4x 3 ! 2x + 6!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!b)!! y = x 5 + x 3 ! x 2 +1
5
3
2
c)!! y = x 4 ! ! 4 !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!d)!! y = ! 3 x 3 ! 3! x
3. Solve:
a)!!if ! p = 4q 3 + 2q 2 ! 5!!!!!!!! find !!
dp
!=
dq
b)!!if !g(t) = 4t 3 ! 3t 2 + 6t!!!!! find !g'(t) =
c)!!if ! y = 2x 7 ! 5x + 3!!!!!!!!!!! find ! y' =
1
4. If y = 2x 3 ! 3x + 7 find:
a)!!! y'!at! x = !2!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!b)!!! y'!at! (1,!5)
c)!!! f (0)!and ! f '(0)
!!!!!
d) How can you use a graphing calculator to check your answers to
these types of questions?
5. Find y' if:
a)!! y = ax 3 + bx 2 + d !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!b)!! y = ax 4 ! ax 2 + bx
c)!! y = 4ax 5 + kx 3 ! Cx + D!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!d)!! y = D 2 x 3 + 5M 3 x 2 ! 7!!!!!!
6. Find
dy
if:
dp
a)!! y = 4 p3 ! 2 p 2 + 6!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!b)!! y = !5p 4 + 6 p !
2
3
c)!! y = 4mp 4 +16 p 2 ! 6c!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!d)!! y = 6a 4 + 8a 3 ! 2 p 2
7. If y = 6x 5 ! 2x 2 + 9x ! 3 find:
a)!! y'!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!b)!! y''!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!c)!! y'''
d)!!
dy
d2y
d 3y
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!e)!! 2 !!!!!!!!!!!!!!!!!!!!!!!!!!!! f )!! 3
dx
dx
dx
2
3
! dy $
! dy $
8
g)!!# & !!!!!!!!!!!!!!!!!!!!!!!!!!!!h)!!# & !!!!!!!!!!!!!!!!!!!!!!!!!i)!! y( )
" dx %
" dx %
2
8. If f (x) = 2x 3 + 6x ! 7 find:
a)!! f '(x)!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!b)!! f ''(x)
2
2
c)!! ( f (x)) !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!d)! ( f '(x)) !
2
(
)
e)!! ( f ''(x)) !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! f )!! f ( ) (x)
9. Find
3
2
dy
if:
dx
a)!! y = Ax 3 ! Bx 2 ! C!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!b)!! y = 5Ax 3 ! 6Bx 2 ! Cx!!
!!!!!!!!!!!!!!!
c)!! y = 5A 2 x 2 ! 6B 4 x 2 ! C 5 x
10. If y = 2Dx 5 ! 3k 2 x 3 + 2 find:
a)!!
dy
dy
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!b)!!
dx
dD
c)!!
dy
dy
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!d)!!
dk
dz
3
Part B:
1. Simplify each expression then find y' .
a)!! y = ( 2x +1) (3x ! 5) !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!b)!! y = ( 2x ! 3)
2
3
c)!! y = ( 4x ) !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!d)! y = x 2 ( x 3 ! 6 )
3
e)!! y = (! x ) ! 3! x!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! f )!! y =
x 2 ! 5x + 4
x !1
2. Rewrite each rational expression using exponents to remove quotients and then find
the first derivative.
5
6
a)!! y = 2 !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!b)!! y = ! 3
x
x
c)!! y =
2
3
5
2
3
!!! 2 !+! !!7x!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!d)!! y = 4x 3 !!! 2 !+!7x !5 !!! !4
4
x
x
x
x
x
3. Rewrite each expression using exponents to remove radicals and quotients, then find
the first derivative.
a)!! y = 10 x !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!b)!! y = 4 3 x 2
c)!! y =
e)!! y =
60
5
x
2
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!d)!! y = x !+!6 x 3
6
7
2
7
2x !3
!+! 3 !!!
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!e)!! y = 5 !!!
4 3
3
x
x
x
x
4
4. Use the PRODUCT RULE to find the first derivative. DO NOT simplify answer.
a)!! y = (3x +1)(2x ! 5)!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!b)!! y = 3x 2 (8x ! 3)
c)!! y = (2x +1)(4x 2 ! 4x +1)!!!!!!!!!!!!!!!!!!!!!!!!!!!!!d)!! y = (3x 3 ! 2x 2 )(3x 3 + 2x 2 )
5. Find
dy
at the given value of x. Do NOT simplify before evaluating.
dx
a)!! y = (2 + 7x)(x ! 3)!;! x = 2!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!b)!! y = (1+ 2x)(1! 2x)!;! x =
1
2
c)!! y = (x 4 ! 4)(x 4 + 4)!;! x = 1
6. Use the QUOTIENT RULE to find the first derivative. DO NOT simplify answer.
x2
4x 2
a)!! y =
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!b)!! y =
2x +1
1! 6x 3
c)!! y =
x 2 ! 4x
x2 ! 9
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!d)!! y = 2
x+2
x +9
e)!! y =
x3
4 ! x2
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
f
)!!
y
=
8 ! x3
3x
5
7. Find
dy
at the given value of x.
dx
a)!! y =
x +1
x 2 !1
!!!,!!
x
=
0!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!b)!!
y
=
!!!,!! x = 1
2x 2 !1
x 2 +1
c)!! y =
x3
2 + x2
!!!,!!
x
=
!1!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!d)!!
y
=
!!!,!! x = !2
8 ! x3
3x
6
Part C:
1. Use the CHAIN RULE to find the first derivative. DO NOT simplify answers.
5
5
a)!! y = ( 6x 2 ) !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!b)!! y = (!3x 4 ) + 6x 2 ! 7x
4
3
c)!! y = ( p 2 ! 3p +1) !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!d)!! y = ( x 2 !1) ( 2x !1)
e)!! y =
1
6x
(x
2
4
+1)
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! f )!! y = ( 2x 4 + 8) 2
4
1
g)!! y = (3t 4 ! 2t ) 4 !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!h)!! y = 5x + 7
1%
"
i)!! y =
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! j)!! y = $1+ u 3 '
2
#
&
4+t
1
6
"
1 %
k)!! y = 1+ u !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!l)!! y = $1+ 3 '
#
x&
(
)
3
6
6
4
3
m)!! y = (! x ) + 2! 2 x + 6! x!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!n)! y = ( 2x 3 + x ) !
o)!! y =
6! x
(x
3
!!)
2
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! p)!! y = 4x 2 ( 2x ! 5)
3
7
2. Find the first derivative of each expression below. DO NOT simplify your answer.
3
a)!! y = ! x + ( 5! x ) !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!b)!! y = (1! x + 2x 2 ! 3x 3 )
3 2
c)!! y = ((2x) + (16 ! x)
4
)
4
2
(2x !1)
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!d)!! y =
3
( x ! 2)
!x
!3
e)!! y = ( 2x !1) !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! f )!! y =
!!)
2
" x 2 !1 %
g)!! y = x (1! 2x ) !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!h)!! y = $ 2 '
# x +1 &
2
(x
3
5
3. Differentiate each expression below. DO NOT simplify.
"! %
3
a)!! f (x) = ! x ! ( 2! x ) !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!b)!!g(x) = ! x + $ '
#x&
c)!!h(x) = ( 2x ! 3x + 5)
2
!1
2
"
1 %
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!d)!! f (x) = $ x!!! x !3 '
#
3 &
!3
4
e)!!k(x) = ! ( x 2 + ! 2 ) !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! f )!! p(x) = ! ( 2x 2 ! 3x +1)
3x
dy
find
by:
x+2
dx
a) using the Quotient Rule
b) using the Product Rule
!4
4. Given y =
c) show the results in (a) and (b)
are identical.
8
5. Find y' if:
a)!! y =
6. If y =
(2x ! 3)
5
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!b)!! y =
( 1! x !) ( x +1)
2
2
3x !1
dy
find
at (-2, -7)
x+3
dx
dy
. DO NOT simplify.
dx
1
1
a)!! y = 2
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!b)!! y =
2
x +6
(5x + 6)
7. Rewrite each quotient as a power and find
c)!! y =
3
(2x
3
! 5)
4
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!d)!! y =
3x
(2x
2
! 5)
4
9
Part D
1. If
a)
b)
c)
y = 2x 3 + 5x ! 7 find:
the rate of change of y with respect to x.
the rate of change of y with respect to x at x = 1.
the rate of change of y with respect to x at (2, 19).
2. If y = 4x 2 ! 6x ! 3 find:
a) y'
dy
b)
dx
c) an expression that calculates the slope of the tangent line at any point.
d) the slope of the tangent line at x = 1
e) the slope of the tangent line at (2, -1)
10
Part E
1. Use IMPLICIT DIFFERENTIATION to find
dy
in terms of x and y.
dx
a)!!4x 2 + y 2 = 8!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!b)!!!3x ! 4y 2 = 2
c)!! x 2 + 5y 2 + y = 10!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!d)!!! xy 2 = 4
e)!! x 2 + 2xy ! y 2 = 13!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! f )!!! y 3 + y = 4x
g)!! y(x 2 + 3) = y 4 +1!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!h)!!! xy 3 + x 3 y = 2
11
2. Write each function in an implicit form without radicals. Differentiate this expression
dy
to find
:
dx
a)!! y = 2 x !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!b)!! y = 3 x
c)!! y = 3! x !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!d)!! y =
3. If x 2 + y 2 = 25 find
3
x
dy
at x = 3
dx
12
Part F:
1. Each position function below describes motion in a straight line. Find the velocity and
acceleration as functions of time (t).
1
a)!!s(t) = 5t 2 ! 2t + 7!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!b)!!s(t) = 4t 4 ! t 2 + 3
2
c)!!s(t) = 6t ! 8!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!d)!!s(t) = t ! 8 +
2
e)!!s(t) = t (t ! 3) !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! f )!!s(t) = t +
6
t
4t
t+2
Part G:
1. If x 2 + y 2 = 8 and
dx
dy
= 3 , find
at (-2, 2).
dt
dt
2. If x 2 + y 2 = z 2 and
dx
dy
dz
= !2 ,
= !1 , x = 1 and y = -3, find
.
dt
dt
dt
13
3. If A is the area of a circle of radius r, find
dA
dr
in terms of
.
dt
dt
4. The area of a circular oil slick on the surface of the sea is increasing at the rate of
150 m2/s. How fast is the radius changing when:
a) the radius 25 m.
b) the area is 1000 m2
5. How fast is the side of a square shrinking when the length of the side is 2 m and the
area is decreasing at 0.25 m2/s ?.
6. The hypotenuse of a right triangle is of fixed length but the lengths of the other two
dx
= 4 and x = 8 if the
sides x and y depends on time. How fast is y changing when
dt
length of the hypotenuse is 17?
7. A spherical balloon is inflated so that the volume is increasing at the rate of 5 m3/min.
a) at what rate is the diameter increasing when the radius is 6 m?
b) at what rate is the diameter increasing when the volume is 36 m3?
14
8. Two cars approach an intersection, one traveling east and the other north. If both cars
are traveling at 70 km/h, how fast are they approaching each other when they are both
0.5 km from the intersection?
Part H:
1. Find
dy
:
dx
!
!$
a)!! y = 3cos 4x!!!!!!!!!!!!!!!!!!!!!!!!b)!! y = cos # 3x + &!!!!!!!!!!!!!!!!!!!!!!!!!!c)!! y = cos ( 2x 3 )
"
2%
d)!! y = cos3 2x!!!!!!!!!!!!!!!!!!!!!!!!e)!! y = cos ( x 2 + x ) !!!!!!!!!!!!!!!!!!!!!!!!!!! f )!! y = ( x + cos x )
2
g)!! y = 2sin ! x + x 2 !!!!!!!!!!!!!!!!h)!! y = 3sin ( x 2 '1) !!!!!!!!!!!!!!!!!!!!!!!!!!i)!! y = (sin 2x + cos x )
2
2. Differentiate each functions:
a)!! f (x) = x cos x!!!!!!!!!!!!!!!!!!!!!!!!!!!!!b)!!g(x) = x 3 sin 2x!!!!!!!!!!!!!!!!!!!!!!!!c)!!k(x) = x 3 cos(3x 2 )
d)!!h(x) = sin ( cos ! x ) !!!!!!!!!!!!!!!!!!!!!e)!!m(x) = sin x cos x!!!!!!!!!!!!!!!!!!!!!! f )!! p(x) =
sin 2x
cos2x
1
2
3
1
g)!!g(x) = sin (x )!!!!!!!!!!!!!!!!!!!!!!!!!!h)!!k(x) = sin !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!i)!! f (x) = ( x 2 + cos2 x )
x
2
15
3. Find
dy
in each case where A, B, m and n are constants:
dx
a)!! y = cos ( Ax + B) !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!b)!! y = A cosn Bx
c)!! y = sin m (x n )!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!d)!! y = Ax n sin m Bx
4. Find
dy
in each case:
dx
a)!! y = 2 tan x ! tan 2x!!!!!!!!!!!!!!!!!!!b)!! y = 3sec ( 2x 2 +1) !!!!!!!!!!!!!!!!!!!c)!! y = 3sec 5x
d)!! y =!! x 2 + sec 2 x !!!!!!!!!!!!!!!!!!!!!e)!! y =
x2
!!!!!!!!!!!!!!!!!!!!!!!!!!! f )!! y = tan(x 2 ) ! tan 2 x
tan x
"1%
g)!! y = x csc x !!!!!!!!!!!!!!!!!!!!!!!!!!h)!! y = x 2 tan $ '!!!!!!!!!!!!!!!!!!!!!!!!i)!! y = sin ( tan x )
#x&
5. Use the derivatives of sin x and cos x to verify the derivatives of cot x and csc x as
given on the Calculus 12 Formula Sheet.
16
6. Find
dy
in each case. Watch for the need for Implicit Differentiation!
dx
a)!! y = cot 2x + csc 2x!!!!!!!!!!!!!!!!!!!!!b)!! y = 2x 3 cot x!!!!!!!!!!!!!!!!!!c)!! y = ( x + csc x 2 )
d)!! y = ! 2 + csc 2 x !!!!!!!!!!!!!!!!!!!!!!!e)!! y =
cot x
!!!!!!!!!!!!!!!!! f )!! y = x !csc x
1+ csc 2 x
g)!! y = sin ( xy) !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!h)!! y = cot ( x + y) !!!!!!!!!!!!!!!!!i)!! y = ( cot x + sin x )
2
Part I:
1. Find the derivative of each function:
a)!! f (x) = ln(x ! 2)!!!!!!!!!!!!!!!!!!!!!!b)!!g(x) = 3ln(4 ! 3x)!!!!!!!!!!!!!!!!!!!!!c)!!k(x) = ln(x 2 + 5)
"
1 %
2
d)!!h(x) = ln x 2 + ln 5!!!!!!!!!!!!!!!!!!!e)!! p(x) = ln $ x +
'!!!!!!!!!!!!!!!!!!! f )!!h(x) = x ln x
#
x&
4
g)!! f (x) = ( ln x ) !!!!!!!!!!!!!!!!!!!!!!!!!h)!! f (x) = ln(x 4 )!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!i)!!g(x) = ( x ln x )
4
3
j)!!h(x) = ( ln x + x ) !!!!!!!!!!!!!!!!!!!!k)!!m(x) = (sin x ) ( ln x ) !!!!!!!!!!!!!!!!!!!!!l)!! p(x) = ln (sin x )
m)!! f (x) =
ln x
!!!!!!!!!!!!!!!!!!!!!!n)!!w(x) = ln (sin x + cos x ) !!!!!!!!!!!!!!!o)!!r(x) = cos2 ( ln x ) !!!!!!!!!
3+ ln x
17
2. By differentiating and simplifying, show that each pair of functions has the same
derivative:
a)!! f (x) = ln (10x 2 ) !!!!!!!!!!!!!!!!!!!and !!!!!!!!!!!!!!!!!!!!!g(x) = ln ( x 2 )
b)!! f (x) = ln (sin 2x ) !!!!!!!!!!!!!!!!!!!and !!!!!!!!!!!!!!!!!!!!!g(x) = ln (sin x ) + ln ( cos x )
c)!! f (x) = ln ( tan x ) !!!!!!!!!!!!!!!!!!!and !!!!!!!!!!!!!!!!!!!!!g(x) = ln (sin x ) ! ln ( cos x )
3. Find
dy
:
dx
a)!! y = x ! !!!!!!!!!!!!!!!!!!!!!!!!!!!!!b)!! y = e x !!!!!!!!!!!!!!!!!!!!!!!!!!!!!c)!! y = e 2
d)!! y = e! !!!!!!!!!!!!!!!!!!!!!!!!!!!!!e)!! y = e! x !!!!!!!!!!!!!!!!!!!!!!!!!!!! f )!! y = xe x
18
4. Differentiate each functions:
a)!! f (x) = 5e 2 x !!!!!!!!!!!!!!!!!!!!!!!!!!!b)!!h(x) = 2e x
2
!x
!!!!!!!!!!!!!!!!!!!!!!!!!!c)!!k(x) = 3e 2sin x
d)!! p(x) = x 2 e x !!!!!!!!!!!!!!!!!!!!!!!!!!!e)!!q(x) = x 2 e!3x !!!!!!!!!!!!!!!!!!!!!!!!!! f )!!m(x) = ( e 2 x ! e!2 x )
g)!!g(x) = x !e x !!!!!!!!!!!!!!!!!!!!!!h)!! f (x) = ln (! + e 2 x ) !!!!!!!!!!!!!!!!!!!!!i)!!m(x) =
e2 x
1+ e 2 x
j)!!g(x) = e x ln x !!!!!!!!!!!!!!!!!!!!!!!!!!!!k)!!w(x) = ln ( e x + e! x ) !!!!!!!!!!!!!!!!!!!!!!!!!!l)!!r(x) =
5. If y defined implicitly as a function of x by the given equation, find
2
ex
ln x
dy
:
dx
a)!! x + ylnx = 2!!!!!!!!!!!!!!!!!!!!!!!!!b)!! y ! e xy = 5!!!!!!!!!!!!!!!!!!!!!!!c)!!esin 2 y + 2x = 4y
19
6. Find
dy
:
dx
a)!! y = x ! !!!!!!!!!!!!!!!!!!!!!!!!!!!b)!! y = ! x !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!c)!! y = e! x
d)!! y = 2 x !!!!!!!!!!!!!!!!!!!!!!!!!!!e)!! y = e x ! x e !!!!!!!!!!!!!!!!!!!!!!!!!! f )!! y = x !
2
2
2
g)!! y = 10 x !!!!!!!!!!!!!!!!!!!!!!!!!h)!! y = 2 sin x !!!!!!!!!!!!!!!!!!!!!!!!!!!!!i)!! y = 3x +3x
7. Find
dy
:
dx
a)!! y = ln x 2 !1 !!!!!!!!!!!!!!!!!!!!!!!!b)!! y = ln x 3 ! 7x +1 !!!!!!!!!!!!!!!!!!!!!!c)!!! y = ( ln x )
3
d)!! y = ln tan x !!!!!!!!!!!!!!!!!!!!!!!e)!! y = cos x ln cos x !!!!!!!!!!!!!!!!!!!!!!!!! f )!! y = sin ( ln x )
Part J:
1. Find
dy
:
dx
3
a)!! y = 5 (3x !1) !!!!!!!!!!!!!!!!!!!!!b)!! y = Ax 3 ! 2Bx 2 + C 4 !!!!!!!!!!!!!!!!!!c)!! y = 5x 3 !
2
6
+ !5
2
x x
20
1. Con’t Find
d)!! y = 10 x !
2
dy
dx
5
3
x2
x2 ! 2
!
!!!!!!!!!!!!!!!!!!!!!!e)!! y =
!!!!!!!!!!!!!!!!!!!!!!!!! f )!!! y = 2
!!!
3x +1
x +2
x 3 x2
3 !5
2
3
2
3
1
2
g)!! y = (x ! 3x ) !!!!!!!!!!!!!!!!!!!!!!!!h)!!! y = (x !1) (2x + 5) !!!!!!!!!!!!!!!!!!!!!!!!i)!! y = !(4x ! 2x) !!
j)!! y =
1
4 ! x2
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!k)!! y = (! x)3 ! (4! x)2 !!!!!!!!!!!!!!!!!!!!!!!!l)! y =!
(2x !1)2
!!!!!!!!!!!!!!!
(x + 3)!3
m)!! y = cos(2x 3 + 5)!!!!!!!!!!!!!!!!!!!!!!!!!n)!! y = sin 5 (3x + 4x 2 )!!!!!!!!!!!!!!!!!!!!!!!!!o)!! y = 4x cos3x
p)!! y = sin
2
x2
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!q)!!
y
=
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!r)!! y = sin 3 (tan 4x)
2
x
3tan 4x
s)!!4x 2 + y 2 = 16!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!t)!! xy 2 = 18!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!u)!! y = x 2 ln(2x + 5)
v)!! y = cos(xy)!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!w)!! y 2 + e x = y!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! y)!! y = 5 x
!!!!
2
21