Calculus
So, you think you can take the derivative? (practice #1)
Example1: If y = x 18 ! x 2 , then y ! = ?
!2x
(A)
18 ! x 2
9! x2
(B)
18 ! x 2
18 ! 3x 2
(C)
2 18 ! x 2
18 ! 2x 2
(D)
18 ! x 2
Example2: If y =
sin x
, then
1+ cos x
dy
dx
=?
1
1+ cos x
1
(B)
1! cos x
1
(C)
1+ sin x
1
(D)
1+ sec2 x
(A)
(Q1.) Find the velocity function of a moving particle that has position s(t ) = t 3 cos(2t )
(A) v (t ) = t 3 ! 2 sin(2t )
(B) v (t ) = !2t 3 sin(2t ) + 3t 2 cos(2t )
(C) v (t ) = !6t 2 sin(2t )
(D) v (t ) = !t 3 cos(2t ) + 3t 2 sin(2t )
(E) v (t ) = !t 3 sin(2t ) ! 3t 2 sin(2t )
1
1+ cos x
dy
=?
, then
sin x
dx
(A) csc x cot x + csc2 x
(B) !csc x tan x + sec2 x
(C) !csc x cot x ! csc2 x
1! sin x
(D)
cos2 x
1! sin x
(E)
sin2 x
(Q2.) If y =
(Q3.)
( x )= ?
d
dx
(A)
3
2
3 x
2
(B)
2
33 x
(C)
(D)
3
(E)
23 x
3
2x
1
3
2x
(Q4.) Which of the following is y !!! , the 3rd derivative of y = e x ?
ex
(A)
ex
8
1
(B)
(C)
(D)
8 ex
8
ex
(Q5.) 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
2
(Q6.) 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
(Q7.)
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 )
( )
dy
dx
5
2
10x sec (x )tan(x 2 )
5x sec 4 (x 2 )tan(x 2 )
sec(x 2 )tan(x 2 )
5 sec 4 (2x )tan(2x )
5 sec5 (2x )tan(2x )
(Q8.) If y = sec5 x 2 , find
(A)
(B)
(C)
(D)
(E)
3
(Q9.)
d !# xe x $&
&
#
dx #" 1+ e x &&%
(A)
(B)
(C)
(D)
(E)
xe x + 2e 2 x
(1+ e x )2
e x + e 2 x ! xe 2 x
(1+ e x )2
xe x + 2e 2 x ! 2xe 2 x
(1+ e x )2
xe x + e x
(1+ e x )2
xe x + e x + e 2 x
(1+ e x )2
x2
, then f !(x ) = ?
(1+ 5x )2
2 x + 8x 2
(1 + 5x )4
x
1 + 5x
x
2(1 + 5x )3
2x
(1 + 5x )3
(Q10.) If f (x ) =
(A)
(B)
(C)
(D)
(E)
10x 2
(1 + 5x )3
Answers: (ex1) D (ex2) A
(Q1.) B (Q2.) C (Q3.) B (Q4.) B (Q5.) C (Q6.) E (Q7.) E (Q8.) A (Q9.) E (Q10.) D
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