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JEE • MATHEMATICS • DIFFERENTIATION

Differentiation practice · Mixed difficulty

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271 Easy
271 of 932
Application +4 | -1
If y=asin⁡mx+bcos⁡mxy=a \sin m x+b \cos m x, then d2ydx2\frac{d^2 y}{d x^2} is equal to
View answer and explanation
Correct option: B
−m2y-m^2 y
Option B (−m2y-m^2 y) is correct because differentiating the given equation twice leads to this result.
272 Medium
272 of 932
Application +4 | -1
ddx[tan⁡−1(a−x1+ax)]\frac{d}{{dx}}\left[ {{{\tan }^{ - 1}}\left( {\frac{{a - x}}{{1 + ax}}} \right)} \right] is equal to
View answer and explanation
Correct option: A
−11+x2- \frac{1}{{1 + {x^2}}}
Option A (−11+x2\frac{{-1}}{{1 + x^2}}) is correct because the expression inside the derivative simplifies to tan⁡−1(a)−tan⁡−1(x)\tan^{-1}(a) - \tan^{-1}(x), and the derivative of tan⁡−1(x)\tan^{-1}(x) is 11+x2\frac{1}{{1 + x^2}}.
273 Easy
273 of 932
Application +4 | -1
Let f(x)=sin⁡x,g(x)=x2f(x)=\sin x, g(x)=x^2 and h(x)=log⁡exh(x)=\log _e x. If F(x)=(hogof⁡)(x)F(x)=(\operatorname{hog of})(x), then F′′(x)F^{\prime \prime}(x) is equal to
View answer and explanation
Correct option: D
−2cosec⁡2x-2 \operatorname{cosec}^2 x
Option D (−2cosec⁡2x-2 \operatorname{cosec}^2 x) is correct because it is the second derivative of the composite function F(x)=(h∘g∘f)(x)F(x) = (h \circ g \circ f)(x).
274 Easy
274 of 932
Analyzing +4 | -1
If y=In(xa+bx)x,y = \mathrm{In}{\left( {\frac{x}{{a + bx}}} \right)^x}, then x3d2ydx2{x^3}\frac{{{d^2}y}}{{d{x^2}}} is equal to
View answer and explanation
Correct option: D
(xdydx−y)2{\left( {x\frac{{dy}}{{dx}} - y} \right)^2}
The correct answer is D ((xdydx−y)2{\left( {x\frac{{dy}}{{dx}} - y} \right)^2}) because it is derived through logarithmic differentiation and algebraic manipulation.
275 Medium
275 of 932
Application +4 | -1
If f(x)={π2[x]−x5},1<x<2f\left( x \right) = \left\{ {\frac{\pi }{2}\left[ x \right] - {x^5}} \right\},1 < x < 2 and [.] denotes the greatest integer function, then f′(π25)f'\left( {\sqrt[5]{{\frac{\pi }{2}}}} \right) is equal to
View answer and explanation
Correct option: C
−5(π/2)45- 5{\left( {\pi /2} \right)^{\frac{4}{5}}}
Option C (−5(π/2)45- 5{\left( {\pi /2} \right)^{\frac{4}{5}}}) is correct because after evaluating the derivative at the given point, the result is −5(π/2)45- 5{\left( {\pi /2} \right)^{\frac{4}{5}}}.
276 Medium
276 of 932
Analyzing +4 | -1
If y=f(x)y = f\left( x \right) and ycos⁡x+xcos⁡y=πy\cos x + x\cos y = \mathrm{\pi }, then the value of f′′(0)f''\left( 0 \right) is
View answer and explanation
Correct option: A
π\pi
Option A (π\pi) is correct because, after implicit differentiation and substitution, the value of f′′(0)f''(0) is found to be equal to π\pi.
277 Easy
277 of 932
Application +4 | -1
If y=∑r=1xtan⁡−1(11+r+r2)y=\sum\limits_{r=1}^x \tan ^{-1}\left(\frac{1}{1+r+r^2}\right), then dydx\frac{d y}{d x} is equal to
View answer and explanation
Correct option: B
11+(1+x)2\frac{1}{1+(1+x)^2}
Option B (11+(1+x)2\frac{1}{1+(1+x)^2}) is correct because the summation simplifies to a telescoping series, and the derivative is then straightforward.
278 Easy
278 of 932
Application +4 | -1
The value of aa in order that f(x)=3sin⁡x−cos⁡x−2ax+bf(x)=\sqrt{3} \sin x-\cos x-2 a x+b decreases for all real value of xx, is given by
View answer and explanation
Correct option: B
a≥1a \geq 1
Option B (a≥1a \geq 1) is correct because for the function to decrease for all real values of x, its derivative must be negative, leading to the inequality a≥1a \geq 1.
279 Easy
279 of 932
Application +4 | -1
Derivative of the function f(x)=log⁡5(log⁡7x),x>7f(x)=\log _{5}\left(\log _{7} x\right), x > 7 is
View answer and explanation
Correct option: A
1x(ln⁡5)(ln⁡7)(log⁡7x)\frac{1}{x(\ln 5)(\ln 7)\left(\log _{7} x\right)}
Option A (1x(ln⁡5)(ln⁡7)(log⁡7x)\frac{1}{x(\ln 5)(\ln 7)(\log _{7} x)}) is correct because it is the result of applying the chain rule and change of base formula for logarithms.
280 Easy
280 of 932
Analyzing +4 | -1
The second order derivative of asin⁡3ta \sin ^3 t with respect to a cos⁡3t\cos ^3 t at t=π/4t=\pi / 4, is
View answer and explanation
Correct option: A
423a\frac{4 \sqrt{2}}{3 a}
Option A (423a\frac{4 \sqrt{2}}{3 a}) is correct because after finding the first and second derivatives and substituting t = π/4\pi/4, the result is 423a\frac{4 \sqrt{2}}{3 a}.
281 Easy
281 of 932
Application +4 | -1
If y=xnlog⁡x+x(log⁡x)n,y = {x^n}\log x + x{\left( {\log x} \right)^n}, then dydx\frac{{dy}}{{dx}} is equal to
View answer and explanation
Correct option: A
xn−1(1+nlog⁡x)+(log⁡x)n−1[n+log⁡x]{x^{n - 1}}\left( {1 + n\log x} \right) + {(\log x)^{n - 1}}[n + \log x]
The correct answer is A. The derivative is found by applying the product rule and chain rule to the given function.
282 Easy
282 of 932
Application +4 | -1
Let gg be the inverse function of differentiable function ff and f′(x)=x101+x2.f'(x) = \frac{{{x^{10}}}}{{1 + {x^2}}}. If g(2)=a,  g(2) = a,\,\, then g′(2)g'(2) is equal to
View answer and explanation
Correct option: B
1+a2a10\frac{{1 + {a^2}}}{{{a^{10}}}}
Option B (1+a2a10\frac{{1 + {a^2}}}{{{a^{10}}}}\) is correct because it is the value of the derivative of the inverse function g at 2, calculated using the derivative of the original function f.
283 Medium
283 of 932
Application +4 | -1
If y=ex+e−xex−e−xy = \frac{{{e^x} + {e^{ - x}}}}{{{e^x} - {e^{ - x}}}}, then dydx\frac{{dy}}{{dx}} is equal to
View answer and explanation
Correct option: D
−cosech2x- \mathrm{cosec}{\mathrm{h}^2}x
Option D (−cosech2x- \mathrm{cosec}{\mathrm{h}^2}x) is correct because the derivative of coth⁡x\coth x is −cosech2x- \mathrm{cosec}{\mathrm{h}^2}x.
284 Easy
284 of 932
Application +4 | -1
If y=log⁡tan⁡xy=\log \sqrt{\tan x}, then the value of dydx\frac{d y}{d x} at x=π4x=\frac{\pi}{4}, is given by
View answer and explanation
Correct option: B
1
Option B (1) is correct because substituting x=π4\frac{\pi}{4} into the derivative of the function yields 1.
285 Medium
285 of 932
Analyzing +4 | -1
Let f(x+y)=f(x)f(y)f\left( {x + y} \right) = f\left( x \right)f\left( y \right) and f(x)=1+(sin⁡2x)g(x)f\left( x \right) = 1 + \left( {\sin 2x} \right)g\left( x \right) where g(x)g\left( x \right) is continuous. Then, f′(x)f'\left( x \right) equals
View answer and explanation
Correct option: B
2f(x)g(0)2f\left( x \right)g\left( 0 \right)
Option B (2f(x)g(0)2f\left( x \right)g\left( 0 \right)) is correct because it is the result of applying the limit definition of the derivative and using the given functional equation.

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286 Medium
286 of 932
Analyzing +4 | -1
If d2xdy2(dydx)3+d2ydx2=k\frac{{{d^2}x}}{{d{y^2}}}{\left( {\frac{{dy}}{{dx}}} \right)^3} + \frac{{{d^2}y}}{{d{x^2}}} = k, then kk is equal to
View answer and explanation
Correct option: A
0
Option A (0) is correct because the given expression simplifies to zero using the relationship between first and second derivatives.
287 Medium
287 of 932
Application +4 | -1
If f′(x)=sin⁡(log⁡x)f'\left( x \right) = \sin (\log x) and y=f(2x+33−2x)y = f\left( {\frac{{2x + 3}}{{3 - 2x}}} \right), then dydx\frac{{dy}}{{dx}} at x=1x = 1 is equal to
View answer and explanation
Correct option: C
12sin⁡log⁡(5)12\sin \log \left( 5 \right)
Option C (12sin⁡log⁡(5)12\sin \log ( 5 )) is correct because it is the result of applying the chain rule and evaluating the derivative at x=1.
288 Easy
288 of 932
Application +4 | -1
If f(x)=xnf(x)=x^n and f′(1)=10f^{\prime}(1)=10, then the value of &#39;nn&#39; is
View answer and explanation
Correct option: A
10
Option A (10) is correct because the value of 'n' is 10, derived from the derivative of the function at x=1.
289 Easy
289 of 932
Understanding +4 | -1
Which one of the following is not true always?
View answer and explanation
Correct option: B
If f(x)f\left( x \right) is continuous at x=ax = a, then it is differentiable at x=ax = a
Option B (If f(x)f(x) is continuous at x=ax = a, then it is differentiable at x=ax = a) is correct because continuity does not guarantee differentiability.
290 Easy
290 of 932
Application +4 | -1
If y=sin⁡−1(5x+121−x213),y = {\sin ^{ - 1}}\left( {\frac{{5x + 12\sqrt {1 - {x^2}} }}{{13}}} \right), then dydx\frac{{dy}}{{dx}} is equal to
View answer and explanation
Correct option: B
11−x2\frac{1}{{\sqrt {1 - {x^2}} }}
Option B (11−x2\frac{1}{{\sqrt {1 - {x^2}} }}) is correct because after substituting x=sin⁡θx = \sin \theta and simplifying, the derivative is 11−x2\frac{1}{{\sqrt {1 - {x^2}} }}.
291 Easy
291 of 932
Application +4 | -1
A rectangle with perimeter 32 cm32 \mathrm{~cm} has greatest area if its length is
View answer and explanation
Correct option: C
8
Option C (8) is correct because a rectangle with a fixed perimeter has the greatest area when it is a square.
292 Easy
292 of 932
Analyzing +4 | -1
If y=sec⁡−1x+1x−1+sin⁡−1x−1x+1y = {\sec ^{ - 1}}\frac{{x + 1}}{{x - 1}} + {\sin ^{ - 1}}\frac{{x - 1}}{{x + 1}}, then dydx\frac{{dy}}{{dx}} is
View answer and explanation
Correct option: B
0
Option B (0) is correct because the expression simplifies to a constant, and the derivative of a constant is zero.
293 Easy
293 of 932
Application +4 | -1
If y=tan⁡−1(cos⁡x1+sin⁡x)y = {\tan ^{ - 1}}\left( {\frac{{\cos x}}{{1 + \sin x}}} \right), then dydx\frac{{dy}}{{dx}} is equal to
View answer and explanation
Correct option: D
−12- \frac{1}{2}
Option D is correct because the expression can be simplified using trigonometric identities, leading to a simple derivative.
294 Medium
294 of 932
Application +4 | -1
If x=etsin⁡t,y=etcos⁡tx = {e^t}\sin t,y = {e^t}\cos t, then d2ydx2\frac{{{d^2}y}}{{d{x^2}}} at x=πx = \pi, is
View answer and explanation
Correct option: D
2eπ\frac{2}{{{e^\pi }}}
Option D (2eπ\frac{2}{{{e^\pi }}}) is correct because after finding the first and second derivatives and substituting t=πt = \pi, we get d2ydx2=2eπ\frac{{{d^2}y}}{{d{x^2}}} = \frac{2}{{{e^\pi }}}.
295 Easy
295 of 932
Application +4 | -1
Let ff be a one-one function satisfying f′(x)=f(x)f^{\prime}(x)=f(x) then (f−1)′′(x)\left(f^{-1}\right)^{\prime \prime}(x) is equal to
View answer and explanation
Correct option: B
−1/x2-1 / x^2
Option B (−1/x2-1 / x^2) is correct because it is the second derivative of the inverse function, calculated using the chain rule and the given condition.
296 Easy
296 of 932
Application +4 | -1
ddx1−sin⁡2x1+sin⁡2x\frac{d}{{dx}}\sqrt {\frac{{1 - \sin 2x}}{{1 + \sin 2x}}} is equal to
View answer and explanation
Correct option: B
−sec⁡2(π4−x)- {\sec ^2}\left( {\frac{\pi }{4} - x} \right)
Option B ( −sec⁡2(π4−x)\ - {\sec ^2}\left( {\frac{\pi }{4} - x} \right)) is correct because it is the correct derivative of the simplified expression after using trigonometric identities.
297 Medium
297 of 932
Application +4 | -1
If f(x)=log⁡x(log⁡ex),f\left( x \right) = {\log _x}({\log _e}x), then f′(x)f'\left( x \right) at x=ex = e is equal to
View answer and explanation
Correct option: D
1e\frac{1}{e}
Option D (1e\frac{1}{e}) is correct because after differentiating the given function and substituting x = e, the result simplifies to 1e\frac{1}{e}.
298 Easy
298 of 932
Application +4 | -1
Let f(x)=∣x∣f(x)=|x| and g(x)=[x]g(x)=[x] where [.] denotes the greatest integer function. Then, (f∘g)′(−2)(f \circ g)^{\prime}(-2), is
View answer and explanation
Correct option: D
none of these
The correct answer is D (none of these) because the function f o g is not differentiable at x = -2.
299 Medium
299 of 932
Application +4 | -1
If y=log⁡e(log⁡ex)y=\log _e\left(\log _e x\right), then eydydxe^y \frac{d y}{d x} is equal to
View answer and explanation
Correct option: B
1x\frac{1}{x}
Option B (1/x) is correct because after applying the chain rule and simplifying, the expression e^y * dy/dx simplifies to 1/x.
300 Medium
300 of 932
Application +4 | -1
The derivative of 1+1x1−1x\frac{1+\frac{1}{x}}{1-\frac{1}{x}} is&nbsp;
View answer and explanation
Correct option: B
−2(1−x)2\frac{-2}{(1-x)^2}
Option B (−2(1−x)2\frac{-2}{(1-x)^2}) is correct because simplifying the function and then differentiating yields this result.
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