Calculus
So, you think you can take the derivative?
(Q1.) Which of the following is y (5) , the 5th derivative of y = e 2 x ?
(G) 32e 2 x
(L) 10e 2 x
(U) 5e 2 x
(C) 2e 2 x
(K) e 32 x
d !# 1 $&
(Q2.)
&= ?
#
d ! #" csc ! &%
!sin!
(A)
cos2 !
!1
(B)
csc ! cot !
!1
(C)
cos ! sin!
(D) cos !
(E) ! cos !
d " 4 x 2 + 5x ! 6 %
(Q3.)
'& = ?
dx $#
x +2
(A) 1
(B) 4
x +1
(x + 2)2
8x ! 3
(D)
(x + 2)2
(C)
!x 2 ! 5x + 6
(E)
(x + 2)2
(Q4.) If y = e x (x ! 1)2 (x + 2)4 , then y ! = ?
(
(A) e x (x ! 1)(x + 2)3 x 2 + 7 x ! 2
(B) 6x e (x ! 1)(x + 2)
x
)
3
(
(C) (x + 1)(x + 2)3 x 2 ! 5x + 2
)
2
4 %
"
(D) e x (x + 1)2 (x + 2)4 $ e x +
+
#
x ! 1 x + 2 '&
2
4 %
"
(E) (x + 1)2 (x + 2)3 $ e x +
+
#
x ! 1 x + 2 '&
1
(Q5.) If f (x ) = 5sin x , then f !(x ) = ?
(A) 5sin x ! ln5 ! cos x
(B) !5cos x " ln5 " sin x
(C) 5sin x ! cos x
(D) 5sin x + cos x + ln5
(E) !5sin x " cos x
d 3 2
x =?
(Q6.)
dx
(A) 3 2 x
( )
(B)
2
33 x
(C)
(D)
3
(E)
23 x
3
2x
1
3
2x
(Q7.) If y = sec3 (ln x ) , then which of the following could be
(A)
(B)
(C)
(D)
dy
?
dx
3 sec3 (ln x ) tan(ln x )
x
2 1
3 sec ( x ) tan( x1 )
ln x
2
3 sec (ln x )
x
3 sec3 ( x1 ) tan( x1 )
(E) 3 ln x sec2 ( x1 ) tan( x1 )
(Q8.) If y = tan!1 x ! 1 , then which of the following could be
(A)
2
x x!1
tan!1 x ! 1
(B)
2(1 + x 2 )
1
(C)
2
2(1 + x ) x ! 1
(D)
(E)
x!1
2(1 + x 2 )
1
2x x ! 1
2
dy
?
dx
(Q9.) If f (x ) = x 8 ! x 2 , then f !(x ) = ?
8 ! 2x 2
(A)
8 ! x2
8
(B)
8 ! x2
1 ! 2x 2
(C)
2 8 ! x2
!x 2
(D)
8 ! x2
8 + x ! x2
(E)
2 8 ! x2
(Q10.) If 2 cos x sin y = 14 , then which of the following could be dy
?
dx
(A) 7 tan x ! 7 tan y
(B) !7 sin x + 7 cos y
(C) tan x tan y
(D) ! sin x cos y
!2 sin x + 2 cos y
(E)
14 cos x + 14 sin y
(Q11.) Find the velocity function of a moving particle that has position s(t ) = t 3 cos t
(A) v (t ) = t 3 ! sint
(B) v (t ) = 3t 2 ! sint
(C) v (t ) = t 3 sint ! 3t cos t
(D) v (t ) = !t 3 sint + 3t 2 cos t
(E) v (t ) = !3t 2 sint
10
dP
(Q12.) If P =
, then
could be?
!t
dt
1 + 2e
20e t
(A)
(2 + e t )2
10e t
(B)
(2 ! e t )2
!20e t
(C)
(1 + e t )2
(D)
20 + e t
(1 + 2e t )2
et
(E)
10 + 2e t
3
(Q13.) Which of the following is the 2nd derivative for y = sin!1 x
!2x
(A)
(1 ! x 2 )3
x
(B)
(1 ! x 2 )3
x
(C)
2 1! x2
1
(D)
(1 ! x 2 )3
(E)
sin!1 x
1! x2
(Q14.) Which of the following is the 2nd derivative for y = ln(sec x + tan x )
(A) tan x + sec x csc x
(B) ln(sec x ) + ln(tan x )
sin x + cos x
(C)
sec x + tan x
(D) sin x cos x
(E) sec x tan x
(Q15.)
d
sin4 ! ! cos 4 !) = ?
(
d!
(A) 0
(B) 8 sin3 ! cos 3 !
(C) 4 sin3 (cos !) + 4 cos 3 (sin !)
(D) 4 sin3 ! cos ! ! 4 cos 3 ! sin !
(E) 4 sin! cos !
(Q16.) If f (x ) = x 2 ln(x )!
(A) 2x ln(x ) ! x
2ln x
(B)
x
(C) 2x ln x
ln(x )
(D)
2x
(E) x 2 ln(x ) ! 21
x2
, then f !(x ) = ?
2
4
x2
(Q17.) If f (x ) =
, then f !(x ) = ?
(1+ 5x )2
(A)
(B)
(C)
(D)
2 x + 8x 2
(1 + 5x )4
x
1 + 5x
x
2(1 + 5x )3
2x
(1 + 5x )3
10x 2
(E)
(1 + 5x )3
(Q18.) If g(x ) = tan(x )! x , then g !(x ) = ?
(A) 1 + sec x
(B) sec2 x ! x
(C) csc2 x
(D) tan2 x
(E) 2 tan x sec2 x
(Q19.) [Lovely Heart Curve] If (x 2 + y 2 ! 1)3 ! x 2 y 3 = 0 , then
(A) 3(2x + 2 y )2 ! 6xy 2
3(2x + 2 y )2 ! 6xy 2
(B) 2
(x + y 2 ! 1)3 ! x 2 y 3
(C)
2xy 3 ! 6x (x 2 + y 2 ! 1)2
6 y (x 2 + y 2 ! 1)2 ! 3x 2 y 2
3xy 3 ! 6x (x 2 + y 2 ! 1)2
(D)
2 y (x 2 + y 2 ! 1)2 ! 6x 2 y 2
2xy 3 + 6x (x 2 + y 2 ! 1)2
(E)
!6 y (x 2 + y 2 ! 1)2 + 3x 2 y 2
5
dy
could be?
dx
d
(ln(ln(ln x ))) = ?
dx
1
(A)
x !ln x !ln(ln x )!ln(ln(ln x ))
x
(B)
ln x !ln(ln x )
ln x
(C)
x !ln(ln x )
ln(ln(ln x ))
(D)
x !ln x !ln(ln x )
1
(E)
x !ln x !ln(ln x )
(Q20.)
r3
(Q21.) If s =
(A)
(Q22.)
r2 + a
r 4 ! 3ar 2
, where a is a constant, then
(r 2 + a)3
(
(B)
2r 4 + 3a
(C)
(r 2 + a)3
ds
=?
dr
2r 4 + 3ar 2
(r 2 + a)3
(D)
r 4 + 3ar
(E)
(r 2 + a)3
2r 4 ! 3ar 2
(r 2 + a)3
)
1
d
26e x = ?
dx
1
(A) !26e
!x
26e x
(B)
x2
!13
(C) x
e
! ex $
dy
(Q23.) If y = tan #
, find
&
dx
" x + 1%
! ex $
! e x $ ! xe x $
(A) sec #
tan #
'
" x + 1&%
" x + 1&% #" (x + 1)2 &%
! e x $ ! xe x $
(B) sec2 #
'
" x + 1&% #" (x + 1)2 &%
! xe x $
(C) sec #
" (x + 1)2 &%
2
! e x $ ! (e x $
(D) sec #
'
" x + 1&% #" (x + 1)2 &%
2
" !e x %
(E) sec $
# (x + 1)2 '&
2
6
(D) 13xe
!1
x2
1
!26e x
(E)
x2