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1/21
Differentiation
DPP-1
First Principle, Basic & Chain Rule
Subjective Section :
1. Find the derivative of the following functions w.r.t. x from the first principle:
(i) ),cos(ln x (ii) xx cos)(sin (iii) Calog where xxa & C is constant
(iv) xsin (v) )x(cos 21 (vi) ).1xsin( 2
(vii) If f(x) = x )1('ffind,xtan 1 . (viii)x
e x
(ix) xesin .
2. Let f, g and h are differentiable functions. If 3)0(;2)0(;1)0( hgf and the derivatives of their pair wise
products at x = 0 are 4)0()'gh(;6)0()'fg( and 5)0()'hf( then compute the value of (fgh)’(0).
3. Find the derivative with respect to x of the function:
2
1xsinxcos
x1
x2arcsin)xcos)(logxsin(log
at
4x
.
4. If 1xxln1xx2
1
2
xy 22
2
prove that 2y = xy’ + ln y’. where’ denotes the derivative.
5. If 323 xKx3)]x(f[ then 0)]x(f[
nx)x("f
5
2
. Find the value of n in terms of K.
6. If
2
xtan
ba
batan
ba
2y 1
22, then show that
22
2
)bcoxa(
xsinb
dx
yd
7. Given )12(cos
)1(3
5 2
2
x
x
xy find
dx
dy
8. Let F(x)=f(x) g(x) h(x) for all real x, where f(x), g(x) and h(x) are differentiable functions. At som point
),x(f4)x('f),x(F21)x('F,x 00000 ).x(kh)x('hand),x(g7)x('g 0000 Then k= …………
9. If f(x)=|x–2| and g(x)=f(f(x)) then g’(x)=…..for x>20.
10. Ifxxx
xxxy
cos103cos55cos
102cos154cos66cos
, then find
dx
dy.
11. If
1
3sinln
2xy , then find
dx
dy.
2/21
Objective Section
1. If |,x|lny then dx
dy
(a) 1/x (b) – 1/x (c) |x|/1 (d) None of these
2. The differential coefficient of )x(logf e with respect to x, when ,xlog)x(f e is
(a) xlog
x
e
(b) x
xloge (c) xlogx
1
e
(d) None of these
3. If )x(lnlog)x(f x , then at x = e, f’(x) equals
(a) – 1/e (b) 1/e (c) – e (d) e
4. If )ee(2
ay a/xa/x and y4
dx
yd
2
2
, then a equals
(a) 2 (b) 1 (c) 1/2 (d) 2/1
5. The derivative of the function ))x2((coscos 2/11 at 6
x
is
(a) 2/1)3/2( (b) 2/1)3/1( (c) 2/13 (d) 2/16
6. d.c. of 202 x.t.r.wxsin is
(a) x
xcosxsin 00
(b) x
x2sin
90
0 (c)
x
x2sin
360
0 (d) None of these
7. If x16cos.x8cos.x4cos.x2cos.xcos)x(f then
4'f is
(a) 2 (b) 2
1 (c) 1 (d) None of these
8. If 1x2)x('f 2 and )x(fy 2 then dx
dy at x = 1 is
(a) 2 (b) 1 (c) – 2 (d) None of these
9. If ,)x(cos)x(siny 211 then dx
dyx
dx
yd)x1(
2
22
(a) 0 (b) 1 (c) 4 (d) 3
10. If )x(fn
1ne)x(f for all Nn and x)x(f0 then )}x(f{dx
dn is equal to
(a) )}x(f{dx
d).x(f 1nn (b) )x(f).x(f 1nn
(c) )x(f).x(f).....x(f).x(f 121nn (d) None of these
11. If dx
dy,10logxlog10logxlogy 10xx10
(a) })10(log1{x
elog 2x
10 (b) })10(log1{x
1 2x
(c) })x(log1{elog
1 210
10
(d) })10(log1{x
0log 2x
10
12. If y = ))(( bxxa - (a - b) tan-1
dx
dythen
bx
xa,
(a) 1 (b) bx
xa
(c) bxxa (d) )[
1
xbxa
3/21
13. If 1
1tan 1
x
xy , then
dx
dy is equal to :
(a) 1x|x|2
1
2
(b)
1xx2
1
2
(c)
1xx2
1
2
(d) None of these
14. If f(x) = ln (ln sin x) then
6'
f has the value :
(a) 2l3
1
n (b)
2l
3
n (c)
2l
3
n (d) None of these
15. Let R)1,1(:f be a differentiable function with f(0)= –1 and f’(0) = 1. Let 2)]2)(2([)( xffxg . Then g’(0)=
(a) 4 (b) –4 (c) 0 (d) –2.
16. If y = sec(tan-1 x), then dx
dy at x = 1 is
(a) 2
1 (b)
2
1 (c) 1 (d) None of these
17. A function f, defined for all positive real numbers, satisfies the equation f(x2) = x3 for every x > 0. Then the value of
f’(4) =
(a) 12 (b) 3 (c) 3/2 (d) Cannot be determined
18. xx ee)x(f - 2 sin x - 3
2x3, then the least value of n for which
0x
n
n
)x(fdx
d
is non – zero is
(a) 5 (b) 6 (c) 7 (d) 8
19. If 1x3x
1xxy
2
24
and
dx
dy = ax + b, then the value of a – b is
(a) 8
cot
(b) 12
5cot
(c)
12
5tan
(d)
8
5tan
20. If f(x) = x4tan (x3) – x ln (1 + x2), then the value of 4
4
dx
))x(f(dat x = 0 is
(a) 0 (b) 6 (c) 12 (d) 24
21. If ,bxa
xlogxy
then
2
23
dx
ydx equals to
(a) ydx
dyx (b)
2
ydx
dyx
(c) x
dx
dyy (d)
2
xdx
dyy
22. If y = logcos
2
eetan
xx1 then y’ (0) is equal to
(a) e + e-1 (b) e – e-1 (c) 2
ee 1 (d) None of these,.
23. If y = eax sin bx, thendx
dya2
dx
yd2
2
is equal to
(a) y)ba( 22 (b) y)ba( 22 (c) –y (d) none of these.
24. Differential coefficient of
n
1
nm
m
x .m
1
ln
nm
x
.nm
1
m
n
x
w.r.t. x is
(a) 1 (b) 0 (c) -1 (d) mnx
4/21
25. If y = f
65
43
x
x& f' (x)= tan x2 then
dx
dy
(a) tan x3 (b) 2
2
)65(
1.
65
43tan2
xx
x
(c) 2
2
2
tan6tan5
4tan3x
x
xf
(d) None
26. If y = log5 log5 x, then dx
dy is equal to
(a) xlogx
1
)5(log
12
(b) )x)(log5(log
1
e
(c) x)5(log
1 (d) none of these.
27. If u = ax + b then ))bax(f(dx
dn
n
is equal to
(a) ))u(f(du
dn
n
(b) ))u(f(du
da
n
n
(c) ))u(f(du
da
n
nn (d) ))u(f(
dx
da
n
nn
28. If xxxy , x > 0, then dy/dx at x = 1 is
(a)
24
243
212
1 (b)
24
243
21
1(c)
28
243 (d)
21
243
DPP-2 (Logarithm)
Subjective Section:
1. If dx
dyfindxxy xx )(ln)(cos ln .
2. If xeexxe exx xeey . Find
dx
dy.
3. If x
yx ya.exlny
find dx
dy.
4. If f(x) =
2cot
21
xxx, then the value of f’(1).
5. If ]a)axln[(xy 11 , prove that 1ydx
dyx
dx
yd)1x(x
2
2
.
6. Find the differential coefficient of the function xlog22x
exsinxsinlog)x(f w.r.t. 1x
7. If 1)cx(
c
)cx)(bx(
bx
)cx)(bx)(ax(
axy
2
, Prove that
xc
c
xb
b
xa
a
x
1
y
'y
8. Let xxsinx )x(taney3
find dx
dy
9. If y = ))((cos)(sin ln bxaeecx xx and a + b = e2
then the value of
dx
dy at x = 1.
10. If f(x) = (x + 1) (x + 2) (x + 3) …. (x + n) men find f’(0). .
5/21
11. If f(x) =
100
1
)101()(n
nnnx then find )101('
)101(
f
f.
12. Let f(x) = h
xhx xhx
h
ln)ln(
0
)(sin))(sin(lim
, then find
2
f .
13. If 2
. xx
exy find y’(1)
14. If x
x xy
x 4
log
4tan2
22
then find
1
xdx
dy.
15. Let y =
x
x
xx
111
1 then find y’(1)
Objective Section:
1. (a) If dx
dy,xy )x( x
(a) )xxlog).ex(logx(y xx (b) )xxlog).ex(logx(y x
(c) )xxlog).ex(logx(y 1xx (d) )xxlog).x(logx(y 1xe
x
2. If k)y(sin)x
2sin(
, then at x = 1, dx
dy equals
(a) (b) 2/ (c) kln)2/( (d) None of these
3. If |xsin||x|)x(f ; then )4/('f equals
(a)
224ln
2
2
4
2/1
(b)
224ln
2
2
4
2/1
(c)
22
4ln
2
2
4
2/1
(d)
22
4ln
2
2
4
2/1
4. If ),nx(f))x(f( n then )x('f
)nx('f
(a) )nx(f
)x(f (b)
)x(f
)nx(f (c) )x(f).nx(f (d) None of these
5. If dx
dythenxy x ,)(sin tan is equal to
(a) )xsinlogxsec1()x(sin 2xtan (b) xcos.)x(sinxtan 1xtan
(c) xsinlogxsec)x(sin 2xtan (d) 1xtan)x(sintan
6. If dx
dythenyxyx nmnm ,)( is
(a) xy
yx (b) xy (c)
y
x (d)
x
y
7. If 0))2tan(log(2)2(sec2
3)(sin 1)2/sin( xxy xx then dy/dx at x = -1 is
(a) 3
32
(b) 3
12
(c) 3
32
(d) 3
32
6/21
DPP-3 (Implicit)
Subjective Section:
1. If .eyx22 yx
yarcsin
22 Prove that 0x,
)yx(
)yx(2
dx
yd3
22
2
2
.
2. If
................x
1x
1x
1xy
, prove that
..............x
1x
1x
x2
1
dx
dy
3. If )yx.(ay1x1 33366 , prove that 6
6
2
2
x1
y1
y
x
dx
dy
.
4. Let
.................x2
1x2
1x2
1x)x(f .
Compute the value of ).100('f).100(f
5. If ,xy24y9yx7x 34224 show that x
y
dx
dy .
6. If ,xsinyxe 2xy then at x=0, dx
dy………
7. If 02 22 byhxyax then prove that x
y
byhx
hyax
dx
dy
.
8. Find
dx
dyif sin y = xx coslogsin .
9. If 2 xy yx then find dx
dy.
10. If 0)1()( xyyxx , then show that 2)1(
1
xdx
dy
.
11. If ................ xxxxyyyy , prove that 1
1
yx
yx
dx
dy.
12. If
........1
cos
1
sin
1
cos
1
sin xxxxy then find y’(0).
13. If
xxxxy find y’.
14. Find ,dx
dy if 1)(tan cot1 xy yx .
15. If x =
toyy
y
y
....
1
1
1, prove that xyyx
dx
dy32 22 .
7/21
Objective Section:
1. If yxy ex then dx
dy
(a) )xln1(
xln
(b)
2)xln1(
xln
(c)
2
2
)xln1(
xln
(d) None of these
2. If .............yxyxy , then dx
dy equals to
(a) 1xy2y2
xy
3
2
(b)
1xy2y2
xy
3
2
(c)
1xy2y2
yx
3
2
(d) None of these
3. If 1yx 22 , then
(a) 01)'y(2"yy 2 (b) 01)'y("yy 2 (c) 01)'y("yy (d) 01)'y(2"yy 2
4. If ,ae)yx( )x/y(tan22 1
a > 0 then )0("y is
(a) 2/e2
a (b) 2/ae (c) 2/ea
2 (d) None of these
5. The value of y” if 05yx5yx2x 223 and y = 1 at x = 1
(a) 27
228 (b)
27
164 (c)
27
166 (d)
27
228
6. If y = y(x) and it follows the relation x cos y + y cos x = π, then y”(0)
(a) 1 (b) – 1 (c) π (d) – π
7. If )yasin(xysin and acosx2x1
A
dx
dy
2 then the value of A is
(a) 2 (b) cos a (c) sin a (d) None of these
8. If sin (x + y) = ex+y –2, thendx
dy is equal to
(a) 1 (b) 2 (c) -1 (d) N.O.T.
9. If x2 + y2 = t
1t and x4 + y4 =
2
2
t
1t , then x3y
dx
dy=
(a) 0 (b) 1 (c) – 1 (d) None of these
10. If
22
221
yx
yxsin = log a, then
dx
dyis equal to
(a) y
x (b)
2x
y (c)
22
22
yx
yx
(d)
x
y
11. If (sin x) (cos y) = ½, then d2y/dx2 at ( )4/,4/ is
(a) – 4 (b) – 2 (c) – 6 (d) 0
12. If x2 + y2 = a2 and k = a
1, then k is equal to
(a) 'y1
"y
(b)
32 )'y1(
|"y|
(c)
2'y1
"y2
(d)
32 )'y1(2
"y
13. Let y be an implicit function of x defined by 01cot22 yxx xx . Then y’(1) equals:
(a) –1 (b) 1 (c) log 2 (d) –log 2
14. If xy2)yxln( , then y’(0) =
(a) 1 (b) – 1 (c) 2 (d) 0
8/21
DPP-4 (Parametric Form & Function of Function)
Subjective Section:
1. Differentiate 22
22
x1x1
x1x1
w.r.t. 4x1 .
2. If ,sineccosy;sineccosx nn then show that 0)4y(ndx
dy)4x( 22
22
3. If ttx 2coscos2 & t2sintsin2y , find the value of )dx/yd( 22 when )2/(t .
4. If x4secy and )t(tanx 1 , prove that .)tt61(
)t1(t16
dx
dy242
4
5. A function y of x is represented parametrically as x = a cos t, y = b sin t Find 2
2
dx
yd.
6. Find 22 /dxyd (independent of t), of the function defined parametrically as x = sin (ln t), y = cos
(ln t).
Objective Section:
1. If ),tcos1(ay),tsint(ax then dx
dy
(a) 2
ttan (b)
2
tcot (c)
2
tsec (d)
2
teccos
2. If 2
2
2 1
3,
1
3
t
ty
t
tx
, then dy/dx at t = 2 is
(a) 3
2 (b)
3
2 (c)
3
4 (d)
3
4
3. If ,cossecy,cossecx 22 then yy’ equals
(a) )x2(x2 2 (b) )x2(x2 2 (c) )2x(x2 2 (d) None of these
4. Let the function )x(fy be given 7t20t5tx 35 and ,3t18t3t4y 23 where ).2,2(t Then )x('f at t = 1 is
(a) 5/2 (b) 2/5 (c) 7/5 (d) None of these
5. The differential coefficient of 1x2
1eccos
2
1
with respect to 2x1 at 2
1x is
(a) – 4 (b) 4 (c) -1 (d) None of these
6. If tlogy,tcosx e then at 2
2
2
dx
dy.
dx
yd,2/t
equals
(a) 1 (b) – 1 (c) 0 (d) 2
7. If bxcos)x('g,axsin)x('f),x(gv),x(fu ba then at x = 1, dv
du equals (a > 0, b > 0)
(a) bsecasinb
a (b) baec
a
bcoscos (c)
a
b (d)
b
a
8. If t1t1
t1t1e x
and
t1
t1
2
ytan
, then
dx
dyat
2
1t is :
(a) – 1/2 (b) 1/2 (c) 0 (d) None of these
9/21
9. Let y = t10 + 1 and x = t8 + 1, then 2
2
dx
yd is
(a) t2
5 (b) 20 t8 (c)
6t16
5 (d) None of these
10. If x = 3 cos t, y = 4 sin t, then dy/dx at the point (x = 23 /2, y = 22 ) is
(a) 3
2 (b)
3
2 (c)
3
4 (d)
3
4
11. Differential coefficient of log10x with respect to logx 10 is
(a) 2
2
)(log
)10(log
x (b)
2
2
)10(log
)10(log x (c) 2
210
)10(log
)(log x (d)
2
2
)10(log
)(log x
12. If a function is represented parametrically by the equations tt
yt
tx
2
2
3,
123
then
(a) dx
dy
dx
dyx
1
2
(b) dx
dy
dx
dyx
1
3
(c) dx
dyx
dx
dy
3
(d) dx
dyx
dx
dyx
1
3
DPP-5 (Inverse) Subjective Section:
1. Let 2x,3x4x)x(f 2 and let g be the inverse of f. Find the value of g’ where f(x) = 2
2. If 2
1
u1
utany
&
1,
2
1
2
1,0u,
1u2
1secx
2
1 prove that 01dx
dy2
3. If ,xsin1xsin1
xsin1xsin1coty 1
find
dx
dy if
,
220x .
4. If
x1
x1tan2sin
x11
xtany 1
2
1 , then find dx
dy for )1,1(x .
5. If ..........13x7x
1tan
7x5x
1tan
3x3x
1tan
1xx
1tany
2
1
2
1
2
1
2
1
to n terms. Find dy/dx, expressing
your answer in 2 terms
6. (a) Find the derivative of
2
21
x1
x1cos when 0x , using the substitution x = tan θ.
(b) If ,x1
x1sin)x(f
2
21
find Rx)x('f , clearly stating the point(s) where f(x) is not derivable . Also draw the graph of
y = f(x) and state its range and monotonic behavior.
7. If the function 2
x2 ex)x(f and g(x)= ),x(f 1 then the value of g’(1) is
8. Let ,2cos
sintansin)(f 1
where .
44
Then the value of ))(f(
)(tand
d
is
9. If y = f(x) = 53 xx and g is the inverse of f then find g’(2) (i.e. dx/dy when y = 2).
10/21
10. If the function 32
14)(32
2
1xx
xexf
x
and g(x) = )(1 xf , then find the value of g’
6
7.
11. Differentiate )1()1(
)1()1(tan 1
xx
xx
.
12. Find the derivative of y =
2
1
1
2sin
x
x and mention the points of non-differentiability. Also find (i)
f’(2) (ii) f’
2
1 (iii) f’(1).
13. Find the derivative of
2
1
1
2tan
x
xy .
14. Find the derivative of y = )34(cos 31 xx .
Objective Section:
1. If
abx
baxtany 1 then
dx
dy
(a) )x1(
1
2 (b)
)x1(
1
(c)
22 )x1(
1
(d) None of these
2. If
1x
x1
21
2tany , then
dx
dy at x = 0 is
(a) 1 (b) 2 (c) 2log)5/3( (d) 10
2log
3. Let ),0[),0[:f be a function defined by 2x)x(fy then
2
2
2
2
dy
xd.
dx
yd is equal to
(a) 1 (b) 2x2/1 (c) 3x2/1 (d) 3x2/1
4. If .xlog61
xlog23tan
)exlog(
)x/elog(tany 1
2
21
then
2
2
dx
yd is
(a) 2 (b) 1 (c) 0 (d) – 1
5. xcoscosdx
d 1 is equal to
(a) xsec12
1 (b) xsec1 (c) xsec1
2
1 (d) xsec1
6.
x1
x1cotsin
dx
d 12 is equal to
(a) – 1 (b) 2
1 (c)
2
1 (d) 1
11/21
7. The function f(x) = ex + x, being differentiable and one to one, has a differentiable inverse f-1(x_. the value of dx
d(f-
1) at the point f(log2) is
(a) 2ln
1 (b)
3
1 (c)
4
1 (d) None of these
8. The derivative of
x
1x1tan
21 with respect to
2
21
x21
x1x2tan at x = 0 is
(a) 1/8 (b) 1/4 (c) 1/2 (d) 1
9.
x31
)x3(xtan
dx
d 1 =
(a) x)x1(2
1
(b)
x)x1(
3
(c)
x)x1(
2
(d)
x)x1(2
3
10. If xexy then
2
2
dy
xd is :
(a) xe (b) 3x
x
e1
e
(c)
2xx
e1
e
(d)
2x)e1(
1
11. If g is the inverse of f & f’(x)=5x1
1
then g’(x)=
(a) 1+[g(x)]5 (b) 5)]x(g[1
1
(c)
5)]x(g[1
1
(d) None
12. 2
2
dy
xdequals
(a)
1
2
2
dx
yd (b)
31
2
2
dx
dy
dx
yd (c)
2
2
2
dx
dy
dx
yd (d)
3
2
2
dx
dy
dx
yd
13. Let f(x) = x + sin x. Suppose g denotes the inverse function of f. The value of g’
2
1
4
has the value
equal to
(a) 12 (b) 2
12 (c) 22 (d) 12
14. If y = cos-1(cos x), then dx
dy at x =
4
5 is
(a) 1 (b) – 1 (c) 2
1 (d) None of these
15. If f(x) = sin-1 cos x, then the value of f(10) + f’(10) is
(a) 2
711
(b) 11
2
7
(c) 11
2
5
(d) None of these
16. If f(x) = 2 sin-1 x1x2sinx1 1 , where
2
1,0x , then f’(x) is
(a) x1x
2
(b) zero (c)
x1x
2
(d)
12/21
17. If
2
1
x1
x2cosy , then
dx
dy is
(a) 2x1
2
for all x (b)
2x1
2
for all |x| < 1 (c)
2x1
2
for |x| > 1 (d) None of these
18. If y = cos-1
13
xsin3xcos2, then
dx
dy is equal to
(a) 1 (b) 0 (c) constant (1) (d) none of these.
DPP-6 (Determinant)
Subjective Section:
1. If
1)cx()cx(
1)bx()bx(
1)ax()ax(
)x(f34
34
34
then
1)cx()cx(
1)bx()bx(
1)ax()ax(
.)x('f24
24
24
. Find the value of λ.
2. If 2
222
222
x2sin0x2sin
)xxsin()xxcos()xxsin(
)xxcos()xxsin()xxcos(
)x(f
then find f’(x).
3. If α be a repeated root of a quadratic equation f(x) = 0 & A(x), B(x), C(x) be the polynomials of degree 3, 4, & 5
respectively, then show that
)('C)('B)('A
)(C)(B)(A
)x(C)x(B)x(A
is divisible by f(x), where dash denotes the derivative.
4. Let .
xrxqxp
xnxmml
xcxbba
)x(f
Show that f”(x) = 0 and that f(x) = f(0) + kx where k denotes the sum of all the co-factors of
the elements in f(0).
5. If Y = sX and Z = tX, where all the letters denotes the functions of x and suffixes denotes the differentiation w.r.t. x then
prove that 22
113
222
111ts
tsX
ZYX
ZYX
ZYX
6. If f, g, h are differentiable functions of x and (x) = ,
)"()"()"(
)'()'()'(222 hxgxfx
xhxgxf
hgf
prove that
)'"()"()'"(
''')(
333 hxgxfx
hgf
hgf
dx
xd
.
7. If f(x) = 32
)2/cos()2/sin(!
cossin
aaa
nnn
xxxn
then find the value of n
n
dx
d (f(x)) at x = 0 for n = 2m + 1.
13/21
Objective Section:
1. If
1x21xx
x3x22x
xlx
432
2
find the value of dx
d at x =0
(a) 2 (b) -2 (c) 4 (d) -4
2. If 32
32
32
)1x()1x(x
)1x(x1x
x)1x(1x
)x(D
the coefficient of x in D(x) is-
(a) 5 (b) -1 (c) 6 (d) 0
3. Let 32
3
016
cossin
)(
ppp
xxx
xf where p is a constant. Then 0))((3
3
xatxfdx
d is
(a) p (b) 2pp (c) 3pp (d) independent of p.
DPP-7 (L-Hospital Rule) Subjective
1.
2
2
10x x
x1
xsinx
1Lim 2.
20x x
)x1ln(xcosxLim
3.
xsin
1
x
1Lim
220x 4. If 1
ax
xaLim
ax
ax
ax
find ‘a’
5. xtan.x
)x1ln(xcosxsin1Lim
20x
6. )x2(tanlogLim 2
xtan0x2
7. Determine the values of a, b and c so that 1x
xsincx)xcosba(Lim
50x
8. )xln(sinxsin1
)x(sinxsinLim
xsin
2x
9. )xcos1)(xsinx(
xx
xsinlnx3
Lim
32
0x
10. )xx2cos(.ln
)x3sin(Lim
2
2
0x
11. If 432
32
0x xx2)x1ln(.x2
xcxbxxsinaLim
exists & is finite, find the values of a, b, c & the limit.
12. Find the values of constants a, b & c so that 2xsinx
cxe)x1ln(baxeLim
2
xx
0x
.
14/21
Objective Section:
1. The value of x
xx
x2sin1
)sin(cos22lim
3
4
is
(a) 2
3 (b)
3
2 (c)
2
1 (d) 2 .
2. ||
2cos2cos
21 xx
xLimitx
=
(a) 2cos 2 (b) -2cos2 (c) 2sin2 (d) -2sin 2
3. If g(x) = - 225 x , then 1
)1()(lim
1
x
gxg
x is equal to
(a) 24
3 (b)
24
1 (c) -
24
1 (d) none of these.
4. If function f(x) is differentiable at x = a then ax
xfaafx
ax
)()(lim
22
is
(a) )(')( 2 afaaaf (b) )(')(2 2 afaaaf (c) )(')(2 2 afaaaf (d) )('2 afa
5. If )(),( agaf nn exist and are not equal for some n. Further if f(a)=g(a)=k and
,4)()(
)()()()()()(lim
xfxg
agxfagafxgaf
ax then the value of k is equal to
(a) 4 (b) 2 (c) 1 (d) 0
6. Let f(2)=4 and f’(2)=4. Then 2
)(2)2(lim
2
x
xfxf
x is given by
(a) 2 (b) –2 (c) –4 (d) 3.
7. For x > 0, xsinx/1
0x)x1()x(sinLim
is
(a) 0 (b) – 1 (c) 1 (d) 2
DPP-8 (Miscellaneous)
Subjective Section:
1. If |,xsin||x4cos|logy u where x2secu , find 6/xdx
dy
2. If f: R → R is a function such that )3('")2(")1(')( 23 fxffxxxf for all ,Rx then prove that
)0(f)1(f)2(f .
3. Show that the substitution
2
xtanlnz changes the equation 0cos4cot 2
2
2
xecydx
dyx
dx
yd to
.0y4)dz/yd( 22
4. Prove that IK,Kx,xsin2
nx2sinx)1n2cos(.........x5cosx3cosxcos and deduce from this:
xsin4
]x)1n2sin()1n2(x)1n2sin()1n2[(x)1n2sin()1n2(.....x5sin5x3sin3xsin
2
5. Find a polynomial function f(x) such that )x("f)x('f)x2(f .
15/21
Objective Section:
1. If |xsin||xcos|y then dx
dy at
3
2x
is
(a) 2
31 (b) 0 (c)
2
13 (d) None of these
2. Let )x(f)x(f , hence f(x) is an even function. Then f’(x) must be
(a) an even function (b) an odd function (c) a periodic function (d) either even nor odd
3. If |x|1
x)x(f
then f’(x) equals
(a) |)x|1(
1 (b)
|x|1
1
(c)
|x|1
x
(d) None of these
4. If }x]x[3
sin{ 2
for 2 < x < 3 and [x] denotes the greatest integer less than or equal to x then )3/('f is
equal to
(a) 3
(b)
3
(c)
2
(d) None of these
5. If a)x(Py2 polynomial of degree 3 then
dx
ydy
dx
d2
23 equals
(a) )x('P)x('"P (b) )x('"p).x("p (c) )x("'P).x(P (d) None of these
6. If 21
3)(5
x
xfxf and )x(xfy then
1xdx
dy
is equal to
(a) 14 (b) 7/8 (c) 1 (d) None of these
7. If )x(g)x('f and )x(f)x('g for all x and f(2) = 4 = f’(2) then )19(g)19(f 22 is
(a) 16 (b) 32 (c) 64 (d) None of these
8. If n)x1()x(f then the value of !n
)0('f........
!2
)0("f)0('f)0(f is
(a) n (b) n2 (c) 1n2 (d) None of these
9. If P(x) be a polynomial of degree 4, with 12)2('"P,2)2("P,0)2('P,1)2(P and 24)2(P iv then P”(1)
is equal to
(a) 22 (b) 24 (c) 26 (d) 28
10. If x3+3x2y-6xy2+2y3=0 then the value of 2
2
dx
yd at (1,1) is:
(a) 1 (b) -1 (c) 6 (d) None of these
11. Assume that h(x) = fog (x), where both f and g are differentiable functions. If 3)1('g.2)1(g and
4)2('f then the value of h’(-1) is
(a) – 6 (b) 6 (c) – 12 (d) 12
12. If ,x2yy m
1
m
1
then the value of y
'xy"y)1x( 2 is equal to value equal to
(a)4m2 (b) 2m2 (c) m2 (d) –m2
16/21
13. Let f(x) be a polynomial function of the second degree. If f(1) = f(-1) and 321 a,a,a are in AP then
)a('f),a('f),a('f 321 are in
(a) AP (b) GP` (c) HP (d) None of these
14. If )x1)........(x1)(x1)(x1)(x1(yn2442 then
dx
dy at x = 0 is
(a) 1 (b) – 1 (c) 0 (d) None of these
15. If xcosxsiny then 17
17
dx
yd equals
(a) xcosxsin (b) xsinxcos (c) xcosxsin (d) xcosxsin
16. If )x1x(y 2n/1 , then 122 xyy)x1( is equal to
(a) yn 2 (b) 2ny (c) 22 yn (d) None of these
17. Suppose f(x) = eax + ebx, where a b, and that f’’(x) – 2f’(x) – 15 f(x) = 0 for all x. Then the product ab is
(a) 25 (b) 9 (c) – 15 (d) – 9
18. A function f satisfies the condition, f(x) = f’(x) + f’’(x) + f’’’(x) + …….. where f(x) is a differentiable
function indefinitely and dash denotes the order of derivative. If f(0) = 1, then f(x) is
(a) ex/2 (b) ex (c) e2x (d) e4x
19. Instead of the usual definition of derivative Df(x), if we define a new kind of derivative, D*F(x) by the
formula D*(x) = h
xfhxf
h
)()(lim
22
0
, where f2(x) means [f(x)]2. If f(x) = x log x, then
D*f(x)|x = e has the value
(a) e (b) 2e (c) 4e (d) None of these
20. If f(x) = |x2 – 5x + 6|, then f’(x) equals
(a) 2x – 5 for 2 < x < 3 (b) 5 – 2x for 2 < x < 3 (c) 2x – 5 for 2 x 3 (d) 5 – 2x for 2 x 3
21. If f(0) = 0, f’(0) – 2, then the derivative of y= f(f(f(f(x)))) at x = 0 is,
(a) 2 (b) 8 (c) 16 (d) 4
22. n
n
dx
d(log x) =
(a) nx
)!1n( (b)
nx
!n (c)
nx
)!2n( (d)
n
1n
x
)!1n()1(
23. Let h(x) be differentiable for all x and let f(x) = (kx + ex) h(x), where k is some constant. If h(0) = 5,
h’(0) = - 2 and f’(0) = 18, then the value of k is
(a) 5 (b) 4 (c) 3 (d) 2.2
17/21
24. 20
20
dx
yd(cosx cos3x) is equal to
(a) 219(cos2x – 220cos 3x) (b) 219 (cos 2x + 220 cos 4x)
(c) 219 (sin 2x + 220 sin 4x) (d) 219 (sin 2x – 220 sin 4x)
25. Suppose the function f(x) – f(2x) has the derivative 5 at x = 1 and derivative 7 at x = 2. The derivative
of the function f(x) - f(4x) at x = 1 ahs the value equal to
(a) 19 (b) 9 (c) 17 (d) 14
26. If f(x) = |loge |x||, then f’(x) equals
(a) ,|x|
1 where x 0 (b)
x
1for |x| > 1 and
x
1 for |x| < 1
(c) x
1 for |x| > 1 and
x
1for |x| < 1 (d)
x
1for x > 0 and
x
1 for x < 0
27. If 2
2
),(),(dx
ydthentytx is
(a) 2)'(
''''''
(b)
3)'(
''''''
(c)
''
''
(d)
''
''
28. If f(x) satisfies the relation
Ryxyfxfyx
f
,
2
)(3)(5
2
35, and f(0) = 3 and f’(0) = 2, then the period of sin(f(x)) is
(a) 2 (b) (c) 3 (d) 4
29. The nth derivative of xex vanishes when
(a) x = 0 (b) x = - 1 (c) x = - n (d) x = n
30. If y2 = ax2 + bx + c, then 2
23
dx
ydy is
(a) a constant (b) a function of x only
(c) a function of y only (d) a function of x and y
31. If 1 is a twice repeated root of the equation ax3 + bx2 + bx + d = 0, then
(a) a = b = d (b) a + b = 0 (c) b + d = 0 (d) a = d
32. If f(x – y), f(x) f(y) and f(x + y) are in A.P. for all x, y, and f(0) 0, then
(a) f(4) = f(-4) (b) f(2) + f(-2) = 0 (c) f’(4) + f’(4) = 0 (d) f’(2) = f’(-2)
33. If F(x)=22
22
xg
xf where f”(x)=–f(x) and g(x)=f’(x) and given that F(5)=5, then F(10) is equal to
(a) 5 (b) 10 (c) 0 (d) 15
18/21
34. Let f(x) be a quadratic expression which is positive for all real values of x if g(x)=f(x)+f’(x)+f”(x), then for any real x
(a) g(x)<0 (b) g(x)>0 (c) g(x)=0 (d) g(x)0
35. Let f and g be differentiable functions such that f (3) = 5. g(3) = 7, f'(3) = 13, g'(3) = 6, f'(7) = 2 and g'(7)
= 0. If h(x) = ( gf ) (x), then h'(3) is equal to
(a) 14 (b) 6 (c) 12 (d) 10
36. Let f (x) = 2/(x + 1) and g(x) = 3x. It is given that )( gf )( 0x = )()( 0xfg . Then )()'(o
xfg equals
(a) 32 (b) 3
32 (c)
9
32 (d)
3
32
37. If f (x) and g(x) are two functions from R to R such that 823 )2()()( xxxgf , then f'(1) g'(1) is
(a) 8 (b) 16 (c) 12 (d) 24
38. If f (x) = x ( 0x ) and g(x) = 21 x , then )1()'( gf is equal to
(a) 1 (b) 2
1 (c)
2
1 (d) 2
39. S1: If f(x) = |2| x , f'(f(x)) = 1 for x > 20
S2: If f(x) = ||1 x
x
, then f'(-1) =
4
1
S3: If f(0) = a, f'(0) = b, g(0) = 0 and (fog)'(0) = c, then g'(0) = b
c
S4: Differential coefficient of x1tan2 w.r.t. 2
1
1
2sin
x
x
at 2
1x is 1
(a) FTTT (b) TFTT (c) TTFF (d) TTTT
40. Let f (x) be a polynomial with positive degree satisfying the relation
f (x) f (y) = f (x)+ f (y) + f (xy) – 2
For all real x and y. suppose f (4) = 65. Then
(a) f'(x) is a polynomial of degree two (b) roots of the equation f' (x) = 2x + 1 are real
(c) xf' (x) = 3[f (x) – 1] (d) f' (-1) = 3
41. Given f(x) = sin3
23
xx
1.5 a – x sin a. sin 2a - )178(sin5 21 aa then:
(a) f'(x) = 8sin4sin6sin22 xx (b) f' (sin 8) > 0
(c) f' (x) is not defined at x = sin 8 (d) f' (sin 8) < 0
42. Let f(x) = x[x], x I, where [.] denotes the greatest integer function, then f’(x) is equal to (a) 2x (b) [x] (c) 2[x] (d) None of these
43. Let f : R R be a differentiable function satisfying f(y) . f(x – y) = f(x) Ryx , and f’(5) = q and
pf
f 1
)0('
)0( then f(5) is
(a) q
p2
(b) q
p (c)
p
q (d) q
44. Let f(x) = Wnxn , . The number of values of n for which f’(p + q) = f’(p) + f’(q) is valid for all positive
p and q is (a) 0 (b) 1 (c) 2 (d) none
19/21
Answer Key (Differentiation)
DPP – 1
Subjective Section
1. (i) x
xlnsin (ii) xxxxx
xcoscotsinlnsinsin
cos (iii)
2ln
ln1ln
xx
cx
(iv) xx
x
sin4
cos (v)
41
2
x
x
(vi) 1cos2 2 xx (vii)
2
1
4
(viii)
2x
exe xx
(ix) .cossin xe x
2. 16
3. 2ln
8
16
322
4. --
5. 2K2n
6. --
7. 1),24sin(2)1(
1.
3
52
xxx
8. k=24
9. 1
10. xsin2
11.
3
2
2
13
sin
13
cot
xIn
xx
Objective Section 1. a 2. c 3. b 4. c 5. c 6. c
7. a 8. a 9. c 10. a,c 11. a 12. b
13. b 14. c 15. b 16. a 17. b 18. c
19. b 20. a 21. b 22. d 23. a 24. b
25. b 26. d 27. c 28. a
DPP – 2
Subjective Section
1.
)ln(ln
ln
1)(ln)(coslntan
)ln(cos)(ln)(cos lnln x
xxxxx
x
xxxDy xxxx
2.
xlne
x
1ex]xlne1[xxexlne
x
ex.e
dx
dy xeex1exxx
ex xxeeexxxe
3. )alnyx1(xln
1yln.xlnxxlnx.
x
y
4. – 1
5. - -
6. )]xln(sin)x(cotxln.x2))xln(sinxcot.xlnx2()x).(sinx([lnxlnx
1x2 222222xln22
2
7. - -
8. –
9. 1sinln
10.
100
1........
4
1
3
1
2
11!100
20/21
11. 5050
1
12. 0
13. 2/5e
14. 4
15. )4ln( e
Objective Section 1. c 2. d 3. d 4. b 5. a 6. d 7. c
DPP – 3 Subjective Section 1. - - 2. - - 3. - - 4. 100 5. – 6. – 7. –
8.
2sinlncos
coslncotsinlntan
xy
xxxx
9.
y
xynxy
x
yxnyy
xy
yx
.
.
10. 21
1
x
11. - - 12. ½
13. xx
xyy
y
ln1
. 1
14.
xyxxn
xyxxecyny
xy
yx
cot.tantan
1.tancos
1cot11
12112cot
15. --
Objective Section 1. b 2. a
3. b 4. c
5. d 6. c
7. c 8. c
9. d 10. d
11. a 12. b
13. a 14. a
DPP – 4 Subjective Section
1. 6
4
x
x11
2. - - 3. 2
3 4. - -
5. ta
b32 sin
1 6.
3
1
y
Objective Section 1. a 2. c
3. a 4. b
5. a 6. c
7. a 8. a
9. c 10. d
11. d 12. b
DPP – 5 Subjective Section 1. 1/6
2. - -
3. 2
1
4. 2x12
x21
5. 22 x1
1
)nx(1
1
6. (a) 2x1
2
; (b) {0}, range
2,
2
7. 2 8. 1
9. 8
1
10. 5
1
11. 212
1
x
21/21
12. (i) f’(2) = - 5
2 (ii) f’
5
8
2
1
13.
1,1,
1,1,1
22
xexistnotdoes
Rxx
14.
2
1||
2
1||
1||2
1
,1
3
,1
3
2
2
x
x
x
x
existnotdoesx
Objective Section 1. a 2. d
3. d 4. c
5. a 6. b
7. b 8. b
9. d 10. b
11. a 12. d
13. c 14. b
15. a 16. b
17. b 18. d
DPP – 6 Subjective Section 1. 3 2. )xx(2cos).x21(2 2 6. 0
Objective Section 1. a 2. a 3. d
DPP – 7 Subjective Section 1.
6
5
2. 2
1
3. 3
1
4. a = 1
5. 2
1
6. 1 7. a = 120; b = 60; c = 180 8. 2 9. – 2/5 10. – 6
11. a = 6, b = 6, c = 0 ; 40
3
12. 9c,12b,3a
Objective Section 1. a 2. c 3. b 4. b 5. a 6. c 7. c
DPP – 8 Subjective Section
1. 4ln
)2ln12(3
dx
dy
2. - - 3. - - 4. - -
5. 9
x4 3
Objective Section 1. c 2. b 3. a 4. d 5. c 6. b 7. b 8. b 9. c
10. d 11. c 12. c 13. a 14. a 15. d 16. a 17. c 18. a
19. c 20. b 21. c 22. d 23. c 24. b 25. a 26. b 27. b
28. b 29. c 30. a 31. c,d 32. a,c 33. a 34. b 35. c 36. d
37. a 38. c 39. d 40. a,b,c,d 41. a,d 42. b 43. c 44. c
