A) Short answer type:
1)
a) If y= cos²x - sin²x,
find y" at x= 0. -4
b) If f(x)= 3sinx - 4sin³x find the value of f"(π/2). 9
c) If y²= mx², prove y"= 0
d) If f(x)= x²eˣ then find f"(0). 2
e) If x= a cos 2t and y= bsin²t, find the value of d²y/dx². 0
f) If y²= 4ax then Prove that
d²y/dx² . d²x/dy² = - 2a/y³
g) If pvᵏ = constant, prove that,
v² d²p/dv² = k(k+1)p
h) sin⁻¹{2x/(1+x²)} w.r.t.tan⁻¹x. 2
I) If xʸ yˣ= c, c being constant, find dy/dx at x= e, y= e. -1
j) If y= tan(sin⁻¹x), find dy/dx when x= √3/2. 8
k) y=sin(log x) find dy/dx at x=1 1
l) If sinx= y sin(x+π/4), then dy/dx = K cosec²(x + π/4) find K. 1/√2
m) eʸ=x, then dy/dx
A) logx. B) xˣ. C) log(ex). D) none
n) y= (e²ˣ -1)/(e²ˣ+1), then show that dy/dx= 1 - y²
o) dy/dx of sinx= 2t/(1+t²) , and tany= 2t/(1-t²). 1
p) If x/(x-y)= log{a/(x-y)}, then dy/dx= 2 - f(x,y). Find f(x,y). x/y
q) If (x³+y³)/(x³-y³)= sec⁻¹a³, find d²y/dx². 0
r) If f(x)= eˣ, g(x)= e⁻ˣ and F(x)= f(g(x)), find dF/dx at x= 0. -e
s) If y= sin⁻¹(2x-4x³), find dy/dx
3/√(1-x²)
t) If y= cos (2 sin⁻¹(cosx)), find dy/dx. 2 sin2x
u) If y= xᵉ eˣ show, x dy/dx=(x+e)y
v) If y=cosecx+ cotx, then show that d²y/dx² = sinx/(1- cosx)²
w) If y= x - x²/2 + x³/3 - x⁴/4+.... To infinity, prove dy/dx= 1/(1+x)
x) If f(x)=logₓ(logₑx) find f'(x) at x=e 1/e
z) If y= sin⁻¹ x satisfy the Equation (1-x²)y"= f(x). y', find f(x) x
a) If aˣ + aʸ = aˣ⁺ʸ, show that
aˣ⁻ʸ. dy/dx +1= 0
b) If y= 1+ x/1! + x²/2! +.....to inf. Find d²y/dx². eˣ
c) If y= x/(x+a), show xy'+y(y-1)=0
d) dy/dx of sin(cosx) w.r.t. cosx
cos(cosx)
e) If y= logₓe , then dy/dx at x= e
A) 1. B) 1/e. C) -1/e. D) none
f) If x= coal - 2cos³k, y= 3sink-2sin³k, then dy/dx is
A) cotk B)tank. C) seck.D)coseck
g) If y= sin⁻¹(3t- 4t³), x= cos⁻¹(1-2t²), show that dy/dx is independent of t.
h) y= tan⁻¹{x³/² - x¹/²)/(1+x²), find dy/dx at x= 2. √2/12
I) If y= tan⁻¹{(1+ log x)/(1- log x)} + tan⁻¹{(1- log x)/(1+log x)} , then dy/dx is equal to
A) 1. B) 0. C) 1/x. D) none
j) If f(x)= tan⁻¹[{√(1+x²) -1}/x] , then f'(0) is equal to
A) 0. B) 1. C) 1/2. D) none
k) If cosec x= -2y log sinx, Prove that, dy/dx + y cot x= 2y² cosx
l) sin⁻¹{t/√(t² +1)} w.r.t.cos⁻¹{1/√(t²+1)}. 1
m) If y= log xˣ, then dy/dx is equal to log (ex)
n) If y= 4 cos³x - 3 cosx find d²y/dx² at x= 0. -9
o) If y= sin⁻¹(cosx), find dy/dx. -1
p) If xʸ = yˣ, find dy/dx. y²(logx -1)/x²(log y-1)
q) tan⁻¹{2x/(1-x²)} w.r.t cos⁻¹{(1-x²)/(1+x²)}. 1
r) xeˣʸ = y+ sin²x then find dy/dx at x=0 1
B) Essay type Questions:
1) xᵃ yᵇ = (x+y)ᵃ⁺ᵇ, then prove that dy/dx is independent of a and b. Hence show that d²y/dx² = 0
2) If y= (1+x)ⁿ/(1- x)ⁿ, show that d²y/dx²= 2(n+x)/(1-x²) . dy/dx
3) If y= tan⁻¹(x/y) then evaluate dy/dx and d²y/dx². y/x, 0
4) If log y=sin⁻¹x prove that, (1-x²)d²y/dx² - x dy/dx = y
5) If y= sin(msin⁻¹x) then prove (1- x²)d²y/dx² - x dy/dx + m²y= 0
6) If y= (sin⁻¹x)² + (cos⁻¹x)² Prove (1- x²)y" - x y'= 4
7) If y= a cos(log x)+ b sin(log x), prove that x² y" + y' + y= 0
8) If x² + xy + y² = a², show that, (x+2y)³ d²y/dx² + 6a² = 0
9) If y²= ax² + 2bx+ c, show that d²y/dx²= (b² - ac)/(ax + b)³
10) If y= A(x+ √(x²-1))ⁿ+ B(x - √(x² - 1))ⁿ, prove (1-x²)y" - x y' + n²y=0
11) If x= cos t and y= log t, then prove that at t=π/2, y" + (y')²= 0
12) If 2x= y¹⁾ᵐ + y⁻¹⁾ᵐ prove that (1-x²)d²y/dx² - x dy/dx + m²y = 0
13) If x= sin t and y= sin nt, show (1-x²)d²y/dx² - x dy/dx +n²y= 0
14) If eˣ= eʸ show (x+y)y"+(y')²= 0
15) If eⁿ sin t and y= eⁿcos t, show (x+y)²d²y/dx²= 2(x dy/dx -y)
16) If y= t² and x= cos t (or sin t), show that, (1-x²)y" - xy' = 2
17) If (a +bx)eʸ⁾ˣ = x show that x³ y" = (xy' - y)²
18) y= Aeⁿ + Be ⁻ⁿ and x= sin t prove, (1-x²) d²y/dx² - x dy/dx= y
19) If x= eⁿ and d²y/dx² + p²y= 0, show that, x²y" + x y' + p²y= 0
20) If p² = a² cos²k+ b² sin²k, prove, p+ d²p/dk² = a²b²/p³
21) rⁿ= aⁿ cos mk, find the value of (r²+ 2r₁²- r r₂)/(r² +r₁²)³⁾² where dr/dk = r₁ and d²r/dk² = r₂
22) y= ax⁵ + bx⁻⁵, show that x²y"+xy' = 25y
23) If y= sin(2 sin⁻¹x), show that, (1-x³) y" = x y' - 4y
24) If cos x= y cos (a+x), show y"= 2sin a sec²(a+x) tan(a+x)
25) dy/dx of log₇(log₇x)
1/log7 . 1/(x logx)
26) If kx² = y+ √(x²+ y²), prove that x dy/dx= y + √(x²+ y²)
27) If x= 3at/(1+t³), y= 3at²/(1+t³) show y'= {t(2-t³)}/(1-2t³). Also find dy/dx at t= 1/2. 5/4
28) If sinx= x cos(a+y), then show that dy/dx= {cos²(a+y)}/cos a also find value of dy/dx at x= 0
Cos a
29) If sec k cos m = C then find d²m/dk².
Cot m(sec²k - tan²k cosec²m)
30) If y= (sinx- x cos x)/(cosx + x sin x), find dy/dx at x=π/2. 1
31) if y= tan⁻¹{(ax-b)/(bx+a) find dy/dx. 1/(1+x²)
32) If f(x)=x√(x²+a²) + a²log(x+ √(x²+ a²)), show f'(0)= 2a
33) If u= sin⁻¹x and v= x³, show that dv/du= 3√{v(v¹⁾³ -v)}
34) If y= log[{x+(√x²+a²)}/a]², show (x²+a²)y" + xy' = 2
35) find dy/dx,if eʸ= tan(π/4+ x/2)
Secx
36) y= tan⁻¹{x sin a)/(1- xcos a)} find dy/dx. sina/(1-2xcos a+x²)
37) y= tan⁻¹{x /(1+20x²)} show dy/dx=5/(1+25x²) - 4/(1+16x²)
38) y=[tan⁻¹{1+√(1-x⁴)}/x²] find dy/dx. -x/√(1-x⁴)
39) y= 5x/³√(1-x²) + cos²(2x+1), find dy/dx.
5(3-x)/{3(1-x)⁵⁾³}- 2sin (4x+2)
40) y=[sin⁻¹{5x+ 12√(1-x²)}/13], find dy/dx. 1/√(1-x²)
41) If xʸ= eˣ⁻ʸ, find dy/dx at x=e
1/4
42) x= a{cos k+ lo tan(k/2)} and y= a sin k, find dy/dx at k=π/4. 1
43) dy/dx off eˣʸ - 4xy = 4 -y/x
44) If x= sin²t/√cos 2t and y= cos²t/√cos 2t, show y²y" = 1
45) y= (x-a)/2 √(2ax- x²) + a²/2 sin⁻¹{(x-a)/a}. √(2ax - x²)
46) If sin y= x sin(a+y), show that y"=2cosec² a sin³(a+y)cos(a+y)
47) If y= 1/3 log{(x+1)/√(x²-x+1)} + 1/3 tan⁻¹{(2x -1)/√3}, Prove that dy/dx= 1/(x³ +1)
48) If y= (1+sinx-cosx)/(1+sinx+cosx), then show that dy/dx= 1/(1+cosx)
49) If y= sec 4x, then show that dy/dx= {16t(1-t⁴)}/(1-6t²+t⁴)², where t= tanx.
50) If x= sin³t/√cos2t and y= cos³t/√cos2t (0<t<π/4) show that dy/dx= 0 at t= π/6
51) If tan y= log x², show that x²dy/dx +(1+2 sin 2y)(1+cos 2y)= 0.
52) Diff. Of sec⁻¹{(1+x²)/(1-x²)} w.r.t.x tan⁻¹{(3x-x³)/(1-3x²)}. 2/3
53) If y= eᵃˣ cos bx, show that d²y/dx²= (a²+b²)eᵃˣ cos(bx + 2tan⁻¹(b/a))
54) dy/dx of y= xᶜᵒˢ ˣ + sin(log x)
xᶜᵒˢ ˣ((1/x) cos x - sinx log x) + (1/x) cos(log x)
55) y= xˣ log(sin x)ˢᶦⁿ ˣ. xˣ log(sin x)ˢᶦⁿ ˣ. [1+ logx + cotx + cotx/log sinx]
56) If y= sinᵃx prove sin²x y"= (m² cos²x - m)y
57) xˣ + (cos x) ˣ. xˣ(1+ log x) + (cos x) ˣ[ log cos x - x tan x]
58) tan⁻¹{√(1+x²)- √(1-x²)}/{√(1+x²)+√(1-x²)} w.r.t cos⁻¹{(x²). -1/2
59) x= 2cos - cos 2t and y= 2 sin t - sin 2t, find d²y/dx² at t=π/2. -3/2
60) If √(1-x⁴)+ √(1-y⁴)= k(x² - y²), show y'= {x√(1-y⁴)}/{y√(1-x⁴)}
61) If xy= a{y+√(y² - x²)}, show that x³y'= y³ + y²√(y² - x²)
62) y=(2x-3)⁵⁾²/{2x-1)³⁾²(2x-5)¹⁾²} find dy/dx at x=3. 3√15/125
63) y= (a+bx)eᵐˣ, a, b, m are constant, show y"-2my'+m²y= 0
64) If tan⁻{(sinx-xcosx)/ (cosy+xsinx)}, find dy/dx. Also find the value of dy/dx at x=1
x²/(1+x²), 1/2
65) if aʸ⁾ˣ = (x+1)e⁻¹⁾ˣ, show that, x(x+1) d²y/dx² + x dy/dx = y
66) If y= (tan⁻¹x)², show that, (1+x²)²d²y/dx² + 2x(1+x²) dy/dx=2
67) If √x/y + √y/x= 6, then show that, dy/dx= (x-17y)/(17x- y)
68) If y= ax/√(a² + x²), prove that (a²+ x²) d²y/dx² + 3x dy/dx = 0
69) find dy/dx, if xʸ = yˣ= aˣ⁺ʸ
(xʸ(loga -y/x) + yˣlog a/y)/(xʸlogx/a + yˣ(x/y - loga))
70) If f(x)={(a+x)/(1+x}ᵃ⁺¹⁺²ˣ Prove f'(0)=(log a² +(1-a²)/a}aᵃ⁺¹
71) If y= √3(3cosk + cos 3k) and x= √3(3sink + sin 3k), find d²y/dx² at k=π/3. 16√3/9
72) y= 2tan⁻¹x[x/{1+√(1-x²)}] + sin[tan⁻¹√{(1-x)/(1+x)}].
√{(1-x)/(1+x)}
73) If x²/a² + y²/b² = 1, prove that a²y² d²y/dx² + b⁴= 0
74) If y= x√(x²+a²)³+ (3/2) a²x √(x²+a²) +(3/2) a⁴ log{x+√(x²+a²)} find dy/dx. 4√(x²+ a²)³
75) dy/dx of x + √(a²+x²).
{x+√(a²+x²)}/√(a²+x²)
76) y= log[{√(x²+a²)+√(x²+b²)}/{√(x²+a²) - √(x²+b²)}], show that dy/dx= 2xloge/√{√(x²+a²)(x²+b²)}
77) If y= (1/4√2) log(x²+ x√2+1)/(x²- x√2+1) +(1/2√2) tan⁻¹{x√2/(1-x²)}, show that dy/dx= 1/(x⁴+1)
78) y=cot⁻¹[{√(1+sinx)+ √(1-sinx)}/{(1+sinx - √(2-sinx)}]. Find dy/dx. 1/2
79) If y= log y= x then prove that (x+y) d²y/dx² + (dy/dx)² = 0
80) dy/dx of (tan⁻¹x)ʸ + y cot x=1
81) If x= f(t), y= g(t) and d²y/dx²= 0, then show that
dx/dt . d²y/dt²= dy/dt . d²x/dt²
82) If ax²+ 2hxy+ by²+ 2gx + 2fy+ c= 0, then show that y"=(abc+2fgh-af²-bg²-ch²)/(hx+by+f)³
83) If 3kx²= y²(k - x⁶), then show that dy/dx= y³/x³ - 2y/x
84)dy/dx: when √(1-y²)+√(1-t²) = a(y-t) and x=sin⁻¹{t√(1-t)+√t√(1-t²) Express your result as a function of y and t, independent of a. 2√t√(1-y²)/(2√t+√(1+t))
85) If y= f{(2x-1)/(x²+1)} and f'(x)= sinx², find dy/dx. 2(1+x-x²)/(1+x²)² sin[{(2x-1)/(x²+1)}²]
86) If x= sec k - cos k and y= secⁿk - cosⁿk then show that (x²+4) d²y/dx² + x dy/dx = n²y
87) If ky= sin(x+y), prove that d²y/dx² + y(1+ dy/dx)³= 0
88) If y= xⁿ⁻¹ log x then show that, x²y"+ (3-2n)xy' + (n -1)²y= 0
89) If y= a sin(log x)+ b cos(log x) find d²y/dx² when dy/dx= 0.
-2(a+b)
90) y= 1 + a/(x-a) +bx/{(x-a)(x-b)} + cx²/{(x-a)(x-b)(x-c)} show that y'= y/x {a/(a-x) + b/(b-x) + c/(c-x)}
91) dy/dx of tan⁻¹{x/a + tan⁻¹y/x}. 1/8
92) y= ₓyˣ find dy/dx
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