482 + Interview Questions in DIFFERENTIATION IN GENERAL AWARENESS Page 5 InterviewSolution

201.	If `y=cos^(-1)((2x)/(1+x^2)),t h e n(dy)/(dx)i s``(-2)/(1+x^2)`for all `x`(b) `(-2)/(1+x^2)`for all `\|x\|1`(d) none of theseA. `(-2)/(1+x^(2))` for all xB. `(-2)/(1+x^(2))" for all "\|x\|gt1`C. `(2)/(1+x^(2))" for "\|x\|lt1`D. none of these
Answer» Correct Answer - D

Discussion

202.	If `y=x^((logx)^log(logx))` then `(dy)/(dx)=`A. `(y)/(x)((In x^(x-1))"+2 In x In(In x))"`B. `(y)/(x)(log x)^(log (log x))(2 log (log x )+1)`C. `(y)/(" x In x")[(In x )^(2)+2In (In x)]`D. `(y)/(x)(log y)/(log x)[2 log (log x)+1]`
Answer» Correct Answer - B `y=x^((log x )^(log (log x)))` `therefore" "log y = (log x) (log x)^(log (log x))" (1)"` Taking log of both sides, we get log (log y ) = log (log x) + log (log x) log (log x) Differentiating w.r.t. x, we get `(1)/(log y).(1)/(y)(dy)/(dx)=(1)/(x log x )+(2 log (log x))/(log x )(1)/(x)` `=(2log (log x)+1)/(x log x)` `"or "(dy)/(dx)=(y)/(x).(log y)/(log x)(2 log (log x )+1)` Substituting the value of log y from (1), we get `(dy)/(dx)=(y)/(x)(log x)^(log(logx))(2 log (log x)+1)`

Discussion

203.	log(1+ sin2x)
Answer» \(\frac{d}{dx}\)log(1+sin²x) \(=\frac{1}{1+sin^2x}\frac{d}{dx}\)(1+sin²x) (By chain rule and \(\frac{d}{dx}\)log x \(=\frac{1}{x}\)) \(=\frac{1}{1+sin^2x}\)(2 sin x cos x) (∵ \(\frac{d}{dx}\) 1 = 0 and \(\frac{d}{dx}\) sin² x = 2 sin x cos x) \(=\frac{sin^2x}{1+sin^2x}\) (∵ 2 sin x cos x = sin² x) Hence, \(\frac{d}{dx}\)log(1+sin² x) \(=\frac{sin^2x}{1+sin^2x}\)

203.

log(1+ sin2x)

Answer»

\(\frac{d}{dx}\)log(1+sin²x)

\(=\frac{1}{1+sin^2x}\frac{d}{dx}\)(1+sin²x) (By chain rule and \(\frac{d}{dx}\)log x \(=\frac{1}{x}\))

\(=\frac{1}{1+sin^2x}\)(2 sin x cos x) (∵ \(\frac{d}{dx}\) 1 = 0 and \(\frac{d}{dx}\) sin² x = 2 sin x cos x)

\(=\frac{sin^2x}{1+sin^2x}\) (∵ 2 sin x cos x = sin² x)

Hence, \(\frac{d}{dx}\)log(1+sin² x) \(=\frac{sin^2x}{1+sin^2x}\)

Discussion

204.	`(d^2x)/(d y^2)` equalsA. `((d^(2)x)/(dy^(2)))^(-1)`B. `((d^(2)x)/(dy^(2)))((dy)/(dx))^(-3)`C. `((d^(2)x)/(dy^(2)))((dy)/(dx))^(-2)`D. `-((d^(2)x)/(dy^(2)))((dy)/(dx))^(-3)`
Answer» Correct Answer - D We have, `(dx)/(dy)=((dy)/(dx))^(-1)` `implies" "(d)/(dy)((dx)/(dy))=(d)/(dy){((dy)/(dx))^(-1)}` `implies" "(d^(2)x)/(dy^(2))=(d)/(dx){((dy)/(dx))^(-1)}(dx)/(dy)` `implies" "(d^(2)x)/(dy^(2))=-((dy)/(dx))^(-2)(d)/(dx)((dy)/(dx)).(dx)/(dy)` `implies" "(d^(2)x)/(dy^(2))=-((dy)/(dx))^(-3)((d^(2)y)/(dx^(2)))`

Discussion

205.	If `y` is a function of `x and log(x+y)-2xy=0` then the value of `y (0)` is equal to (a) `1` (b) `-1` (c) `2` (d) `0`A. 1B. -1C. 2D. 0
Answer» Correct Answer - A When x=0, we have `log(0+y)-2xx0xxy=0implieslogy=0impliesy=1` Now, `log(x+y)-2xy=0` Differentiating with respect to x, we get `(1)/(x+y)(1+(dy)/(dx))-2y-2x(dy)/(dx)=0` Putting x=0 and y=1, we get `(1+(dy)/(dx))-2-2xx0=0implies(dy)/(dx)=1`

Discussion

206.	Let g(x) be the inverse of an invertible function f(x) which is derivable at x = 3. If f(3) = 9 and f’(3) = 9, write the value of g’(9).
Answer» From the definition of invertible function, g(f(x)) = x …(i) So, g(f(3)) = 3, i.e., g(9) = 3 Now, differentiating both sides of equation (i) w.r.t. x using the Chain Rule of Differentiation, we get – g’(f(x)). f’(x) = 1 …(ii) Plugging in x = 3 in equation (ii) gives us – g’(f(3)).f’(3) = 1 or, g’(9).9 = 1 i.e., g’(9) = 1/9

206.

Let g(x) be the inverse of an invertible function f(x) which is derivable at x = 3. If f(3) = 9 and f’(3) = 9, write the value of g’(9).

Answer»

From the definition of invertible function,

g(f(x)) = x …(i)

So, g(f(3)) = 3, i.e., g(9) = 3

Now, differentiating both sides of equation (i) w.r.t. x using the Chain Rule of Differentiation, we get –

g’(f(x)). f’(x) = 1 …(ii)

Plugging in x = 3 in equation (ii) gives us –

g’(f(3)).f’(3) = 1

or, g’(9).9 = 1

i.e., g’(9) = 1/9

Discussion

207.	If y = sin-1(sin x), -\(\cfrac{\pi}2\) ≤ x ≤ \(\cfrac{\pi}2\). Then write the value of \(\cfrac{dy}{d\mathrm x}\) for x ∈ (-\(\cfrac{\pi}2\), \(\cfrac{\pi}2\)).
Answer» For x ∈ \((-\cfrac{\pi}2,\cfrac{\pi}2),\) y = sin^-1(sin x) = x So, \(\cfrac{dy}{d\mathrm x}\) = 1

207.

If y = sin-1(sin x), -\(\cfrac{\pi}2\) ≤ x ≤ \(\cfrac{\pi}2\). Then write the value of \(\cfrac{dy}{d\mathrm x}\) for x ∈ (-\(\cfrac{\pi}2\), \(\cfrac{\pi}2\)).

Answer»

For x ∈ \((-\cfrac{\pi}2,\cfrac{\pi}2),\)

y = sin^-1(sin x)

= x

So, \(\cfrac{dy}{d\mathrm x}\) = 1

Discussion

208.	`tan(sin^(-1)x)`
Answer» `(1)/((1-x^(2))^(3//3))`

Discussion

209.	`e^(tan^(-1)sqrtx)`
Answer» `(e^(tan^(-1)sqrtx))/(2sqrtx(1+x))`

Discussion

210.	Differentiate with respect to x:tan 3x
Answer» Chain rule: If y = f(t) and t = g(x) then dy/dx = dy/dt x dt/dx Let y = tan 3x If t = 3x then tan 3x = tan t or y = tan t dy/dt = sec² t dt/dx = 3 Now, dy/dx = dy/dt x dt/dx = 3 sec² t Substituting the value of t back, we get dy/dx = 3sec² 3x

210.

Differentiate with respect to x:tan 3x

Answer»

Chain rule:

If y = f(t) and t = g(x) then dy/dx = dy/dt x dt/dx

Let y = tan 3x

If t = 3x then tan 3x = tan t or y = tan t

dy/dt = sec² t

dt/dx = 3

Now,

dy/dx = dy/dt x dt/dx = 3 sec² t

Substituting the value of t back, we get

dy/dx = 3sec² 3x

Discussion

211.	Differentiate with respect to x:sin 4x
Answer» Chain rule: If y = f(t) and t = g(x) then dy/dx = dy/dt x dt/dx Let y = sin 4x If 4x = t then sin4x = sin t or y = sin t dy/dt = cos t dt/dx = 4 Now, dy/dx = dy/dt x dt/dx = 4 cos t Substituting the value of t back, we get dy/dx = 4 cos 4x

211.

Differentiate with respect to x:sin 4x

Answer»

Chain rule:

If y = f(t) and t = g(x) then dy/dx = dy/dt x dt/dx

Let y = sin 4x

If 4x = t then sin4x = sin t or y = sin t

dy/dt = cos t

dt/dx = 4

Now,

dy/dx = dy/dt x dt/dx = 4 cos t

Substituting the value of t back, we get

dy/dx = 4 cos 4x

Discussion

212.	Differentiate with respect to x:cos 5x
Answer» Chain rule: If y = f(t) and t = g(x) then dy/dx = dy/dt x dt/dx Let y = cos 5x If t = 5x then cos 5x = cos t or y = cos t dy/dt = -sin t dt/dx = 5 Now, dy/dx = dy/dt x dt/dx = -5 sin t Substituting the value of t back, we get dy/dx = -5 sin 5x

212.

Differentiate with respect to x:cos 5x

Answer»

Chain rule:

If y = f(t) and t = g(x) then dy/dx = dy/dt x dt/dx

Let y = cos 5x

If t = 5x then cos 5x = cos t or y = cos t

dy/dt = -sin t

dt/dx = 5

Now,

dy/dx = dy/dt x dt/dx = -5 sin t

Substituting the value of t back, we get

dy/dx = -5 sin 5x

Discussion

213.	`tan^(-1)(cos sqrtx)`
Answer» `(-sinsqrtx)/((2sqrtx)(1+cos^(2)sqrtx))`

Discussion

214.	Differentiate with respect to x:cot2 x
Answer» Chain rule: If y = f(t) and t = g(x) then dy/dx = dy/dt x dt/dx Let y = cot² x dy/dx = – 2cot x [d/dx (cotx)] = – 2 cot x cosec²x

214.

Differentiate with respect to x:cot2 x

Answer»

Chain rule:

If y = f(t) and t = g(x) then dy/dx = dy/dt x dt/dx

Let y = cot² x

dy/dx = – 2cot x [d/dx (cotx)]

= – 2 cot x cosec²x

Discussion

215.	Differentiate with respect to x.cos x3
Answer» Chain rule: If y = f(t) and t = g(x) then dy/dx = dy/dt x dt/dx Let y = cos x³ If t = x³ then cos x³= cos t or y = cos t dy/dt = -sin t dt/dx = 3x² Now, dy/dx = dy/dt x dt/dx = 3 x² (-sin t) Substituting the value of t back, we get dy/dx = 3 x² (-sin x³) = -3 x² sin x³

215.

Differentiate with respect to x.cos x3

Answer»

Chain rule:

If y = f(t) and t = g(x) then dy/dx = dy/dt x dt/dx

Let y = cos x³

If t = x³ then cos x³= cos t or y = cos t

dy/dt = -sin t

dt/dx = 3x²

Now,

dy/dx = dy/dt x dt/dx = 3 x² (-sin t)

Substituting the value of t back, we get

dy/dx = 3 x² (-sin x³)

= -3 x² sin x³

Discussion

216.	If `y=e^(sqrt(x))+e^(-sqrt(x))," then "(dy)/(dx)` is equal toA. `(e^(sqrt(x))-e^(-sqrt(x)))/(2sqrt(x))`B. `(e^(sqrt(x))-e^(-sqrt(x)))/(2x)`C. `(1)/(2sqrt(x))sqrt(y^(2)-4)`D. `(1)/(2sqrt(x))sqrt(y^(2)+4)`
Answer» `(dy)/(dx)=(e^(sqrt(x)))/(2sqrt(x))-(e^(-sqrt(x)))/(2sqrt(x))=(e^(sqrt(x)-)e^(-sqrt(x)))/(2sqrt(x))` `=(sqrt((e^(sqrt(x))+e^(-sqrt(x)))^(2))-4)/(2sqrt(x))` `=(sqrt(y^(2)-4))/(2sqrt(x))`

Discussion

217.	`(cot^(-1)x^(2))^(3)`
Answer» `(-6x(cot^(-1)x^(2))^(2))/((1+x^(4)))`

Discussion

218.	`log(sin^(-1)x^(4))`
Answer» `(4x^(3))/((sin^(-1)x^(4))sqrt(1-x^(8)))`

Discussion

219.	`f(x)=e^(x)-e^(-x)-2 sin x -(2)/(3)x^(3).` Then the least value of n for which `(d^(n))/(dx^(n))f(x)\|underset(x=0)` is nonzero isA. 5B. 6C. 7D. 8
Answer» `f(x)=e^(x)-e^(-x)-2 sin x -(2)/(3)x^(3)` `f^(I)(x)=e^(x)+e^(-x)-2 cos x -2x^(2)` `f^(II)(x)=e^(x)-e^(-x)+2 sin x-4x` `f^(III)(x)=e^(x)+e^(-x)+2 cos x -4` `f^(IV)(x)=e^(x)-e^(-x)-2 sin x` `f^(V)(x)=e^(x)+e^(-x)-2 cos x` `f^(VI)(x)=e^(x)-e^(-x)+2 sin x` `f^(VII)(x)=e^(x)+e^(-x)+2 cos x` `"Clearly, "f^(VII)(0)" is nonzero."`

Discussion

220.	`tan^(-1)(cotx)`
Answer» Correct Answer - `-1`

Discussion

221.	`(i)tan (logx)" "(ii)log sec x" "(iii) log sin.(x)/(2)`
Answer» `(i)(sec^(2)(logx))/(x)" "(ii)tanx" "(iii)(1)/(2)cot.(x)/(2)`

Discussion

222.	If `y=e^(sqrt(x))+e^(-sqrt(x))`, then `(dy)/(dx)i se q u a lto``(e^(sqrt(x)))/(2sqrt(x))`(b) `(e^(sqrt(x))-e^(-sqrt(x)))/(2x)``1/(2sqrt(x))sqrt(y^2-4)`(d) `1/(2sqrt(x))sqrt(y^2+4)`A. `(e^(sqrt(x))-e^(-sqrt(x)))/(2sqrt(x))`B. `(e^(sqrt(x))-e^(-sqrt(x)))/(2x)`C. `(1)/(2sqrt(x))sqrt(y^(2)-4)`D. `(1)/(2sqrt(x))sqrt(y^(2)+4)`
Answer» `(dy)/(dx)=(e^(sqrt(x)))/(2sqrt(x))-(e^(-sqrt(x)))/(2sqrt(x))=(e^(sqrt(x)-)e^(-sqrt(x)))/(2sqrt(x))` `=(sqrt((e^(sqrt(x))+e^(-sqrt(x)))^(2))-4)/(2sqrt(x))` `=(sqrt(y^(2)-4))/(2sqrt(x))`

Discussion

223.	`(1+x^(2))tan^(-1)x`
Answer» Correct Answer - `(1+2x tan^(-1)x)`

Discussion

224.	If `y=cos^(-1)((2x)/(1+x^2)),t h e n(dy)/(dx)i s``(-2)/(1+x^2)`for all `x`(b) `(-2)/(1+x^2)`for all `\|x\|1`(d) none of theseA. `(-2)/(1+x^(2))` for all xB. `(-2)/(1+x^(2))" for all "\|x\|lt 1`C. `(2)/(1+x^(2))" for "\|x\|gt1`D. none of these
Answer» `y=cos^(-1)((2x)/(1+x^(2)))` `"or "(dy)/(dx)=(1)/(sqrt(1-(4x^(2))/((1+x^(2))^(2))))(d)/(dx)((2x)/(1+x^(2)))` `=-(1+x^(2))/(sqrt((1+x^(2))^(2)))(2(1+x^(2))-4x^(2))/((1+x^(2))^(2))` `=-2(1+x^(2))/(\|1-x^(2)\|)(1-x^(2))/((1+x^(2))^(2))` `=-2((1-x^(2))/(\|1-x^(2)\|))((1)/(1+x^(2)))` `={{:((2)/(1+x^(2))",",if \|x\|gt1),((-2)/(1+x^(2))",",if\|x\|lt1):}`

Discussion

225.	`cos^(-1)2x`
Answer» `(-2)/(sqrt(1-4x^(2)))`

Discussion

226.	`sin^(-1)(cosx)=`
Answer» Correct Answer - `-1`

Discussion

227.	`(e^(x))/((1+cosx))`
Answer» `(e^(x)(1+cosx+sinx))/((1+cos x)^(2))`

Discussion

228.	`e^(xcosx)`
Answer» `e^(x cos x)(cos x- x sin x)`

Discussion

229.	`tan^(-1)x^(2)`
Answer» Correct Answer - `(2x)/((1+x^(4)))`

Discussion

230.	`cot^(-1)x^(3)`
Answer» `(-3x^(2))/((1+x^(6)))`

Discussion

231.	Differentiate`x^(3)e^(x)cosx` w.r.t x
Answer» `e^(x)x^(2)(x cos x-x sin x+3 cos x)`

Discussion

232.	Find `(dy)/(dx)`, when: `y=x^((cos^(-1)x))`
Answer» Correct Answer - `x^(cos^(-1)x)[(cos^(-1)x)/(x)-(logx)/(sqrt(1-x^(2)))]`

Discussion

233.	`log(tan^(-1)x)`
Answer» `(1)/((1+x^(2))tan^(-1)x)`

Discussion

234.	`sec^(-1)sqrtx`
Answer» Correct Answer - `(1)/(2xsqrtx-1)`

Discussion

235.	Find dy/dx: x2/a2 + y2/b2 = 1
Answer» Given as x²/a² + y²/b² = 1 Differentiating the equation on both sides with respect to x, 2x/a² + (2y/b²)(dy/dx) = 0 dy/dx = -xb²/ya²

235.

Find dy/dx: x2/a2 + y2/b2 = 1

Answer»

Given as x²/a² + y²/b² = 1

Differentiating the equation on both sides with respect to x,

2x/a² + (2y/b²)(dy/dx) = 0

dy/dx = -xb²/ya²

Discussion

236.	`y=tan^(- 1)((2x)/(1+15 x^2))f i n d(dy)/(dx)`
Answer» Correct Answer - `(5)/((1+25x^(2)))-(3)/((1+9x^(2)))` `y=tan^(-1)5x-tan^(-1)3x.`

Discussion

237.	Find `(dy)/(dx)`, when: `y=x^(sinx)`
Answer» Correct Answer - `x^(sinx)[(sinx)/(x)+(cosx)logx]`

Discussion

238.	`"sin"^(-1)(x)/(a)`
Answer» `(1)/(sqrt(a^(2)-x^(2)))`

Discussion

239.	`cot^(-1)(e^(x))`
Answer» `(-e^(x))/((1+e^(2x)))`

Discussion

240.	Find `(dy)/(dx)`, when: `y=x^(1//x)`
Answer» Correct Answer - `(x^(1//x)(1-logx))/(x^(2))`

Discussion

241.	Find `(dy)/(dx)`, when: `y=(logx)^(x)`
Answer» Correct Answer - `(logx)^(x){(1)/(logx)+log(logx)}`

Discussion

242.	`tan^(-1)(logx)`
Answer» `(1)/(x{1+(logx)^(2)})`

Discussion

243.	`(i)log(x+(1)/(x))" "(ii)log sin3x" "(iii)log(x+sqrt(1+x^(2)))`
Answer» `(i)((x^(2)-1))/(x(x^(2)+1))" "(ii)3cot3x" "(iii)(1)/(sqrt(1+x^(2)))`

Discussion

244.	Find `(dy)/(dx)`, when: `y=x^(sqrtx)`
Answer» Correct Answer - `(x^(sqrtx)(2+logx))/(2sqrtx)`

Discussion

245.	`(2x-3x+4)^(5)`
Answer» Correct Answer - `5(2x^(2)-3x+4)^(4)(4x-3)`

Discussion

246.	`e^(sqrtx)log x`
Answer» `(e^(sqrtx)(2+sqrtxlogx))/(2x)`

Discussion

247.	`log sin sqrt(x^(2)+1)`
Answer» `(x)/(sqrt(x^(2)+1))cotsqrt(x^(2)+1)`

Discussion

248.	`e^(3x)cos 2x`
Answer» `e^(3x)(3cos 2x-2 sin 2x)`

Discussion

249.	If `y="sec"(tan^(-1)x),`then `(dy)/(dx)`at `x=1`is equal to:`1/(sqrt(2))`(b) `1/2`(c) 1(d) `sqrt(2)`A. `1//2`B. 1C. `sqrt(2)`D. `1sqrt(2)`
Answer» `y=sec(tan^(-1)x)` `rArr" "(dy)/(dx)=sec (tan^(-1)x)cdottan (tan^(-1)x)cdot(1)/(1+x^(2))` `rArr" "((dy)/(dx))_(x=1)=sec((pi)/(4))tan((pi)/(4))(1)/(1+1)=(1)/(sqrt(2))`

Discussion

250.	If `y = tan^(-1)[(sqrt(1+x)-sqrt(1-x))/(sqrt(1+x)+sqrt(1-x))]` then prove that `dy/dx = 1/(2 sqrt(1-x^2))`
Answer» `sec^(-1)theta=cos^(-1).(1)/(theta)and cos^(-1)theta+sin^(-1)theta=(pi)/(2).`

Discussion

Explore topic-wise InterviewSolutions in .

If `y=cos^(-1)((2x)/(1+x^2)),t h e n(dy)/(dx)i s``(-2)/(1+x^2)`for all `x`(b) `(-2)/(1+x^2)`for all `|x|1`(d) none of theseA. `(-2)/(1+x^(2))` for all xB. `(-2)/(1+x^(2))" for all "|x|gt1`C. `(2)/(1+x^(2))" for "|x|lt1`D. none of these

If `y=x^((logx)^log(logx))` then `(dy)/(dx)=`A. `(y)/(x)((In x^(x-1))"+2 In x In(In x))"`B. `(y)/(x)(log x)^(log (log x))(2 log (log x )+1)`C. `(y)/(" x In x")[(In x )^(2)+2In (In x)]`D. `(y)/(x)(log y)/(log x)[2 log (log x)+1]`

log(1+ sin2x)

`(d^2x)/(d y^2)` equalsA. `((d^(2)x)/(dy^(2)))^(-1)`B. `((d^(2)x)/(dy^(2)))((dy)/(dx))^(-3)`C. `((d^(2)x)/(dy^(2)))((dy)/(dx))^(-2)`D. `-((d^(2)x)/(dy^(2)))((dy)/(dx))^(-3)`

If `y` is a function of `x and log(x+y)-2xy=0` then the value of `y (0)` is equal to (a) `1` (b) `-1` (c) `2` (d) `0`A. 1B. -1C. 2D. 0

Let g(x) be the inverse of an invertible function f(x) which is derivable at x = 3. If f(3) = 9 and f’(3) = 9, write the value of g’(9).

If y = sin-1(sin x), -\(\cfrac{\pi}2\) ≤ x ≤ \(\cfrac{\pi}2\). Then write the value of \(\cfrac{dy}{d\mathrm x}\) for x ∈ (-\(\cfrac{\pi}2\), \(\cfrac{\pi}2\)).

`tan(sin^(-1)x)`

`e^(tan^(-1)sqrtx)`

Differentiate with respect to x:tan 3x

Differentiate with respect to x:sin 4x

Differentiate with respect to x:cos 5x

`tan^(-1)(cos sqrtx)`

Differentiate with respect to x:cot2 x

Differentiate with respect to x.cos x3

If `y=e^(sqrt(x))+e^(-sqrt(x))," then "(dy)/(dx)` is equal toA. `(e^(sqrt(x))-e^(-sqrt(x)))/(2sqrt(x))`B. `(e^(sqrt(x))-e^(-sqrt(x)))/(2x)`C. `(1)/(2sqrt(x))sqrt(y^(2)-4)`D. `(1)/(2sqrt(x))sqrt(y^(2)+4)`

`(cot^(-1)x^(2))^(3)`

`log(sin^(-1)x^(4))`

`f(x)=e^(x)-e^(-x)-2 sin x -(2)/(3)x^(3).` Then the least value of n for which `(d^(n))/(dx^(n))f(x)|underset(x=0)` is nonzero isA. 5B. 6C. 7D. 8

`tan^(-1)(cotx)`

`(i)tan (logx)" "(ii)log sec x" "(iii) log sin.(x)/(2)`

`(1+x^(2))tan^(-1)x`

If `y=cos^(-1)((2x)/(1+x^2)),t h e n(dy)/(dx)i s``(-2)/(1+x^2)`for all `x`(b) `(-2)/(1+x^2)`for all `|x|1`(d) none of theseA. `(-2)/(1+x^(2))` for all xB. `(-2)/(1+x^(2))" for all "|x|lt 1`C. `(2)/(1+x^(2))" for "|x|gt1`D. none of these

`cos^(-1)2x`

`sin^(-1)(cosx)=`

`(e^(x))/((1+cosx))`

`e^(xcosx)`

`tan^(-1)x^(2)`

`cot^(-1)x^(3)`

Differentiate`x^(3)e^(x)cosx` w.r.t x

Find `(dy)/(dx)`, when: `y=x^((cos^(-1)x))`

`log(tan^(-1)x)`

`sec^(-1)sqrtx`

Find dy/dx: x2/a2 + y2/b2 = 1

`y=tan^(- 1)((2x)/(1+15 x^2))f i n d(dy)/(dx)`

Find `(dy)/(dx)`, when: `y=x^(sinx)`

`"sin"^(-1)(x)/(a)`

`cot^(-1)(e^(x))`

Find `(dy)/(dx)`, when: `y=x^(1//x)`

Find `(dy)/(dx)`, when: `y=(logx)^(x)`

`tan^(-1)(logx)`

`(i)log(x+(1)/(x))" "(ii)log sin3x" "(iii)log(x+sqrt(1+x^(2)))`

Find `(dy)/(dx)`, when: `y=x^(sqrtx)`

`(2x-3x+4)^(5)`

`e^(sqrtx)log x`

`log sin sqrt(x^(2)+1)`

`e^(3x)cos 2x`

If `y="sec"(tan^(-1)x),`then `(dy)/(dx)`at `x=1`is equal to:`1/(sqrt(2))`(b) `1/2`(c) 1(d) `sqrt(2)`A. `1//2`B. 1C. `sqrt(2)`D. `1sqrt(2)`

If `y = tan^(-1)[(sqrt(1+x)-sqrt(1-x))/(sqrt(1+x)+sqrt(1-x))]` then prove that `dy/dx = 1/(2 sqrt(1-x^2))`