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JEE • MATHEMATICS • DIFFERENTIATION

Differentiation practice · Easy

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These sets use single-correct-answer MCQs. They do not cover every JEE Main or JEE Advanced question format and are not full-length mock papers.

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61 Easy
61 of 500
Application +4 | -1
The derivative of y=(1−x)(2−x)…(n−x)y = \left( {1 - x} \right)\left( {2 - x} \right) \ldots \left( {n - x} \right) at x=1x = 1 is equal to
View answer and explanation
Correct option: B
(−1)(n−1)!\left( { - 1} \right)\left( {n - 1} \right)!
Option B ((−1)(n−1)!\left( { - 1} \right)\left( {n - 1} \right)!\) is correct because it is the value of the derivative of the given product at x=1x = 1, obtained by using logarithmic differentiation.
62 Easy
62 of 500
Application +4 | -1
If y=log⁡sin⁡x(tan⁡x)y=\log _{\sin x}(\tan x), then dydx\frac{d y}{d x} at x=π4x=\frac{\pi}{4} is equal to
View answer and explanation
Correct option: C
−4log⁡2\frac{-4}{\log 2}
Option C (−4log⁡2\frac{-4}{\log 2}) is correct because it is the result of differentiating the logarithmic function and evaluating at x=π4x=\frac{\pi}{4}.
63 Easy
63 of 500
Application +4 | -1
If y=xx2,y = {x^{{x^2}}}, then dydx\frac{{dy}}{{dx}} equals
View answer and explanation
Correct option: D
xx2+1.  (log⁡(ex2)){x^{{x^2} + 1}}.\,\,(\log (e{x^2}))
Option D (xx2+1.(log⁡(ex2)){x^{{x^2} + 1}}.{\left( {\log (e{x^2})} \right)}) is correct because it is the derivative of the function y = x(x2) calculated using logarithmic differentiation.
64 Easy
64 of 500
Application +4 | -1
If xm⋅yn=(x+y)m+nx^{m} \cdot y^{n}=(x+y)^{m+n}, then dydx\frac{d y}{d x} is
View answer and explanation
Correct option: A
yx\frac{y}{x}
Option A (yx\frac{y}{x}) is correct because after taking the logarithm and differentiating the given equation, the expression simplifies to dydx=yx\frac{dy}{dx} = \frac{y}{x}.
65 Easy
65 of 500
Application +4 | -1
If y=sec⁡(tan⁡−1x)y=\sec \left(\tan ^{-1} x\right), then dydx\frac{d y}{d x} at x=1x=1 is equal to
View answer and explanation
Correct option: A
12\frac{1}{\sqrt{2}}
Option A (12\frac{1}{\sqrt{2}}) is correct because it is the value of the derivative of the given function at the specified point.
66 Easy
66 of 500
Application +4 | -1
Let f:R→Rf: R \rightarrow R be a function satisfying f(x+y)=f(x)+2y2+kxyf(x+y)=f(x)+2 y^2+k x y for all x,y∈Rx, y \in R. If f(1)=2f(1)=2 and f(2)=8f(2)=8, then f(x)f(x) is equal to
View answer and explanation
Correct option: A
2x22 x^2
Option A is correct because using the given condition f(x+y)=f(x)+2y2+kxyf(x+y)=f(x)+2 y^2+k x y and the values f(1)=2f(1)=2 and f(2)=8f(2)=8, we find f(x)=2x2f(x)=2 x^2.
67 Easy
67 of 500
Application +4 | -1
ddx(cos⁡−11+cos⁡x2)\frac{d}{d x}\left(\cos ^{-1} \sqrt{\frac{1+\cos x}{2}}\right) is equal to
View answer and explanation
Correct option: B
12\frac{1}{2}
Option B (12\frac{1}{2}) is correct because the expression simplifies to x/2x/2 and its derivative is 1/21/2.
68 Easy
68 of 500
Application +4 | -1
If y=sin⁡−1(5x+121−x213)y=\sin ^{-1}\left(\frac{5 x+12 \sqrt{1-x^2}}{13}\right), then dydx\frac{d y}{d x} 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 it is the derivative of the given function after simplification using trigonometric substitution.
69 Easy
69 of 500
Analyzing +4 | -1
If ddx(f(x))=g(x);ddx(g(x))=f(x2)\frac{d}{d x}(f(x))=g(x) ; \frac{d}{d x}(g(x))=f\left(x^{2}\right), then d2dx2f(x3)\frac{d^{2}}{d x^{2}} f\left(x^{3}\right) can be expressed in the form kxaf(xb)+pxg(xx)k x^{a} f\left(x^{b}\right)+p x g\left(x^{x}\right) where a,b,c,k,p∈Na, b, c, k, p \in N, then the value of (k+p+a+b+c)(k+p+a+b+c) equal to
View answer and explanation
Correct option: C
28
Option C (28) is correct because after finding the second derivative of f(x3) and comparing it to the given form, the sum of the coefficients and exponents is 28.
70 Easy
70 of 500
Application +4 | -1
Let f(x)=x3+4x2+6xf(x)=x^3+4 x^2+6 x and g(x)g(x) be its inverse then the value of g′(−4)g^{\prime}(-4) :
View answer and explanation
Correct option: C
12\frac{1}{2}
Option C (12\frac{1}{2}) is correct because it is found by using the inverse function derivative formula and evaluating at the correct point.
71 Easy
71 of 500
Analyzing +4 | -1
The derivative of cos⁡−1(1−x21+x2){\cos ^{ - 1}}\left( {\frac{{1 - {x^2}}}{{1 + {x^2}}}} \right) with respect to cot⁡−1(1−3x23x−x3){\cot ^{ - 1}}\left( {\frac{{1 - 3{x^2}}}{{3x - {x^3}}}} \right) is
View answer and explanation
Correct option: C
23\frac{2}{3}
Option C (23\frac{2}{3}) is correct because by using the substitution x=tan⁡θx = \tan \theta, we can simplify the expressions and find that the derivative of y1{y_1} with respect to y2{y_2} is 23\frac{2}{3}.
72 Easy
72 of 500
Application +4 | -1
A spherical balloon is expanding. If the radius is increasing at the rate of 5 inch per minute, the rate at which the volume increases (in cubic inches per minute) when the radius is 10 inch is
View answer and explanation
Correct option: C
2000π2000 \pi
Option C (2000π2000 \pi) is correct because it represents the rate of change of the volume of the sphere when the radius is 10 inches and increasing at 5 inches per minute.
73 Easy
73 of 500
Analyzing +4 | -1
The range of a∈Ra \in R for which the function
f(x)=(4a−3)(x+log⁡e5)+2(a−7)cot⁡(x2)sin⁡2(x2),  x≠2nπ,  n∈Nf\left( x \right) = \left( {4a - 3} \right)\left( {x + {{\log }_e}5} \right) + 2\left( {a - 7} \right)\cot \left( {\frac{x}{2}} \right){\sin ^2}\left( {\frac{x}{2}} \right),\,\,x \ne 2n\pi ,\,\,n \in N has critical points, is:
View answer and explanation
Correct option: A
[−43, 2]\left[ { - \frac{4}{3},\,2} \right]
Option A ([−43, 2]\left[ { - \frac{4}{3},\,2} \right]) is correct because it represents the range of 'a' for which the given function has critical points, derived from the condition -1 <= cos(x) <= 1.
74 Easy
74 of 500
Application +4 | -1
If y=sin⁡x+sin⁡x+sin⁡x+...∞,y = \sqrt {\sin x + \sqrt {\sin x + \sqrt {\sin x + ...\infty ,} } }, then dydx\frac{{dy}}{{dx}} is equal to
View answer and explanation
Correct option: A
cos⁡x2y−1\frac{{\cos x}}{{2y - 1}}
Option A (cos x / (2y - 1)) is correct because squaring the equation and then differentiating implicitly gives cos x / (2y - 1).
75 Easy
75 of 500
Application +4 | -1
If y=cos⁡x2+cos⁡x2+cos⁡x2+⋯+∞y=\sqrt{\cos x^2+\sqrt{\cos x^2+\sqrt{\cos x^2+\cdots+\infty}}}, then dydx\frac{d y}{d x}
View answer and explanation
Correct option: B
−2xsin⁡x22y−1\frac{-2 x \sin x^2}{2 y-1}
Option B (−2xsin⁡x22y−1\frac{-2 x \sin x^2}{2 y-1}) is correct because differentiating the given equation implicitly leads to this result.

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SEQUENCE OF SERIES 1764 MCQs DEFINITE INTEGRATION 1443 MCQs MATRICES 1363 MCQs INDEFINITE INTEGRATION 1179 MCQs PROBABILITY 1146 MCQs COMPLEX NUMBERS 980 MCQs
76 Easy
76 of 500
Application +4 | -1
d2xdy2\frac{d^2 x}{d y^2} equals
View answer and explanation
Correct option: A
−(d2ydx2)−1(dydx)−3-\left(\frac{d^2 y}{d x^2}\right)^{-1}\left(\frac{d y}{d x}\right)^{-3}
Option A (−(d2ydx2)−1(dydx)−3-\left(\frac{{d^2 y}}{{d x^2}}\right)^{-1}\left(\frac{{d y}}{{d x}}\right)^{-3}) is correct because it represents the second derivative of x with respect to y in terms of derivatives of y with respect to x.
77 Easy
77 of 500
Application +4 | -1
If y=log⁡{(1+x1−x)14}−12tan⁡−1xy = \log \left\{ {{{\left( {\frac{{1 + x}}{{1 - x}}} \right)}^{\frac{1}{4}}}} \right\} - \frac{1}{2}{\tan ^{ - 1}}x, then dydx=\frac{{dy}}{{dx}} =
View answer and explanation
Correct option: B
x21−x4\frac{{{x^2}}}{{1 - {x^4}}}
Option B (x21−x4\frac{{{x^2}}}{{1 - {x^4}}}) is correct because it is the simplified derivative after applying logarithmic and inverse trigonometric differentiation rules.
78 Easy
78 of 500
Application +4 | -1
If yy is a function of xx and log⁡(x+y)=2xy\log (x+y)=2 x y, then (dydx)x=0\left(\frac{d y}{d x}\right)_{x=0} is
View answer and explanation
Correct option: A
1
Option A (1) is correct because differentiating the given equation and substituting x=0 yields dy/dx = 1.
79 Easy
79 of 500
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
12sinlog(5)12\mathrm{ }sin\mathrm{ }log\mathrm{ }\left( 5 \right)
Option C (12sinlog(5)12\mathrm{ }sin\mathrm{ }log\mathrm{ }\left( 5 \right)) is correct because it is the value of the derivative of the composite function evaluated at x=1x = 1.
80 Easy
80 of 500
Application +4 | -1
Let solution of differential equation e−x(dy−ℓn(ex)dx)−dx=0\mathrm{e}^{-\mathrm{x}}(\mathrm{dy}-\ell \mathrm{n}(\mathrm{ex}) \mathrm{dx})-\mathrm{dx}=0 is given by y=f(x)\mathrm{y}=\mathrm{f}(\mathrm{x}). If f(1)=e+1\mathrm{f}(1)=\mathrm{e}+1, then value of lim⁡x→0+f(x)\lim \limits_{\mathrm{x} \rightarrow 0^{+}} \mathrm{f}(\mathrm{x}) is
View answer and explanation
Correct option: A
2
Option A (2) is correct. The solution of the differential equation is found, the initial condition is applied to find the constant, and then the limit is evaluated.
81 Easy
81 of 500
Analyzing +4 | -1
x=cosθ,y=sin⁡5θ⇒(1−x2)d2ydx2−xdydxx = \mathrm{cos\theta },y = \sin 5\mathrm{\theta } \Rightarrow \left( {1 - {x^2}} \right)\frac{{{d^2}y}}{{d{x^2}}} - x\frac{{dy}}{{dx}} is
View answer and explanation
Correct option: D
−25y- 25y
Option D (-25y) is correct because after finding the first and second derivatives and substituting, the expression simplifies to -25y.
82 Easy
82 of 500
Application +4 | -1
If y=sin⁡−11−x2y=\sin ^{-1} \sqrt{1-x^2}, then dy/dx=d y / d x= ?
View answer and explanation
Correct option: C
−11−x2-\frac{1}{\sqrt{1-x^2}}
Option C (-1/sqrt(1-x2)) is correct because it is the derivative of the given function after simplification using trigonometric substitution.
83 Easy
83 of 500
Application +4 | -1
If x=3tan⁡tx=3 \tan t and y=3sec⁡ty=3 \sec t, then the value of d2ydx2\frac{d^{2} y}{\mathrm{dx}^{2}} at t=π4\mathrm{t}=\frac{\pi}{4}, is:
View answer and explanation
Correct option: B
162\frac{1}{6 \sqrt{2}}
Option B (162\frac{1}{6 \sqrt{2}}) is correct because it is the value of the second derivative calculated using parametric differentiation at the given point.
84 Easy
84 of 500
Analyzing +4 | -1
If y2=ax2+bx+c{y^2} = a{x^2} + bx + c where a,b,ca,b,c are constants, then y3d2ydx2{y^3}\frac{{{d^2}y}}{{d{x^2}}}, is equal to
View answer and explanation
Correct option: A
A constant
Option A (A constant) is correct because after differentiating the given equation twice and simplifying, the expression y3d2ydx2{y^3}\frac{{{d^2}y}}{{d{x^2}}} results in a constant value.
85 Easy
85 of 500
Understanding +4 | -1
If y=tan⁡−1x+cot⁡−1x+sec⁡−1+cosec−1x,y = {\tan ^{ - 1}}x + {\cot ^{ - 1}}x + {\sec ^{ - 1}} + \mathrm{cose}{\mathrm{c}^{ - 1}}x, then dydx\frac{{dy}}{{dx}} is equal to
View answer and explanation
Correct option: C
0
Option C (0) is correct because the expression simplifies to a constant, and the derivative of a constant is zero.
86 Easy
86 of 500
Application +4 | -1
If y=log⁡xx,y = \log {x^x}, then the value of dydx\frac{{dy}}{{dx}} is
View answer and explanation
Correct option: B
log⁡(ex)\log \left( {ex} \right)
Option B (log(ex)) is correct because simplifying y and then differentiating gives log(ex).
87 Easy
87 of 500
Analyzing +4 | -1
The functions u=exsin⁡x;v=excos⁡xu=e^x \sin x ; v=e^x \cos x satisfy the equation
View answer and explanation
Correct option: D
All the above
Option D (All the above) is correct because the given functions satisfy all the equations provided when their derivatives are calculated and substituted.
88 Easy
88 of 500
Application +4 | -1
The derivative of the function 2x2+6x+12 x^2+6 x+1 at x=3x=3 is
View answer and explanation
Correct option: A
18
Option A (18) is correct because substituting x=3 into the derivative of the function yields 18.
89 Easy
89 of 500
Application +4 | -1
If f′(x)=sin⁡2xf^{\prime}(x)=\sin ^2 x and y=f(2x−1x2+1)y=f\left(\frac{2 x-1}{x^2+1}\right) then dy/dxd y / d x at x=1x=1 is
View answer and explanation
Correct option: D
12sin⁡2(12)\frac{1}{2} \sin ^2\left(\frac{1}{2}\right)
Option D (12sin⁡2(12)\frac{1}{2} \sin ^2(\frac{1}{2})) is correct because, after applying the chain rule and simplifying, we evaluate the derivative at x=1x=1, which gives 12sin⁡2(12)\frac{1}{2} \sin ^2(\frac{1}{2}).
90 Easy
90 of 500
Analyzing +4 | -1
The second order derivative of asin⁡3asin⁡3ta{\sin ^3}a{\sin ^3}t with respect to acos⁡3cos⁡3t a{\cos ^3}{\cos ^3}t\, at t=π4 t = \frac{\pi }{4}\, is
View answer and explanation
Correct option: C
423a\frac{{4\sqrt 2 }}{{3a}}
Option C (423a\frac{{4\sqrt 2 }}{{3a}}) is correct because after finding the second-order derivative and evaluating it at t=π4t = \frac{\pi }{4}, the expression simplifies to the stated result.
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