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JEE • MATHEMATICS • DIFFERENTIATION

Differentiation practice · Mixed difficulty

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841 Medium
841 of 932
Analyzing +4 | -1
If y=tan⁡−1(log⁡(e/x2)log⁡(ex2))+tan⁡−1(3+2log⁡x1−6log⁡x)y = {\tan ^{ - 1}}\left( {\frac{{\log \left( {e/{x^2}} \right)}}{{\log \left( {e{x^2}} \right)}}} \right) + {\tan ^{ - 1}}\left( {\frac{{3 + 2\log x}}{{1 - 6\log x}}} \right), then d2ydx2\frac{{{d^2}y}}{{d{x^2}}} is equal to
View answer and explanation
Correct option: C
0
Option C (0) is correct because after simplification, y becomes a constant, and the second derivative of a constant is zero.
842 Easy
842 of 932
Application +4 | -1
If sin⁡−1[x−ax−a−ax]\sin ^{-1}[\sqrt{x-a x}-\sqrt{a-a x}], then dydx=\frac{d y}{d x}=
View answer and explanation
Correct option: C
12x1−x\frac{1}{2 \sqrt{x} \sqrt{1-x}}
Option C (12x1−x\frac{{1}}{{2 \sqrt{{x}} \sqrt{{1-x}}}}) is correct because it is the derivative of the simplified form of the given function after applying trigonometric substitution.
843 Easy
843 of 932
Analyzing +4 | -1
If y=sin⁡x+exy=\sin x+e^x, then d2xdy2\frac{d^2 x}{d y^2} equals
View answer and explanation
Correct option: C
sin⁡x−ex(cos⁡x+ex)3\frac{\sin x-e^x}{\left(\cos x+e^x\right)^3}
Option C is correct because it involves finding the second derivative of x with respect to y using the chain rule and inverse function differentiation.
844 Hard
844 of 932
Analyzing +4 | -1
If y=(ax+bcx+d),y = \left( {\frac{{ax + b}}{{cx + d}}} \right), then 2dydx.d3ydx32\frac{{dy}}{{dx}}.\frac{{{d^3}y}}{{d{x^3}}} is equal to
View answer and explanation
Correct option: C
3(d2ydx2)23{\left( {\frac{{{d^2}y}}{{d{x^2}}}} \right)^2}
Option C (3(d2ydx2)23{\left( {\frac{{{d^2}y}}{{d{x^2}}}} \right)^2}) is correct because after differentiating the given function three times, the expression 2dydx.d3ydx32\frac{{dy}}{{dx}}.\frac{{{d^3}y}}{{d{x^3}}} simplifies to 3(d2ydx2)23{\left( {\frac{{{d^2}y}}{{d{x^2}}}} \right)^2}.
845 Easy
845 of 932
Application +4 | -1
If f(x)=cos⁡−1{1−(log⁡ex)21+(log⁡ex)2}f(x)=\cos ^{-1}\left\{\frac{1-\left(\log _e x\right)^2}{1+\left(\log _e x\right)^2}\right\}, then f′(1e)f^{\prime}\left(\frac{1}{e}\right) is equal to
View answer and explanation
Correct option: B
−e-e
Option B is correct because it correctly simplifies the inverse cosine function and finds its derivative, then evaluates it at the given point.
846 Easy
846 of 932
Application +4 | -1
If f(x)=x2−xx2+2xf(x)=\frac{x^2-x}{x^2+2 x}, then ddx{f−1(x)}\frac{d}{d x}\left\{f^{-1}(x)\right\} is equal to
View answer and explanation
Correct option: B
3(1−x)2\frac{3}{(1-x)^2}
Option B is correct because it correctly finds the inverse function and then differentiates it.
847 Medium
847 of 932
Application +4 | -1
If y=tan⁡−11+x2−1−x21+x2+1−x2,y = {\tan ^{ - 1}}\frac{{\sqrt {1 + {x^2}} - \sqrt {1 - {x^2}} }}{{\sqrt {1 + {x^2}} + \sqrt {1 - {x^2}} }}, then dydx\frac{{dy}}{{dx}} is equal to
View answer and explanation
Correct option: D
x1−x4\frac{x}{{\sqrt {1 - {x^4}} }}
Option D (x1−x4\frac{x}{{\sqrt {1 - {x^4}} }}) is correct because it is the result of simplifying the expression using trigonometric substitution and then differentiating.
848 Medium
848 of 932
Analyzing +4 | -1
If f(x)=∫0x(f(t))2dt,f:R→Rf(x)=\int_0^x(f(t))^2 d t, f: R \rightarrow R be differentiable function and f(g(x))f(g(x)) is differentiable function at x=ax=a, then
View answer and explanation
Correct option: A
g(x)g(x) must be differentiable at x=ax=a
Option A (0) is correct because, after substitution, the integral becomes the integral of an odd function over a symmetric interval, which is zero.
849 Easy
849 of 932
Analyzing +4 | -1
Derivative of sec⁡−1(11−2x2)w.r.t.sin⁡−1(3x−4x3){\sec ^{ - 1}}\left( {\frac{1}{{1 - 2{x^2}}}} \right)\mathrm{w}\mathrm{.r}\mathrm{.t}\mathrm{.}{\sin ^{ - 1}}\left( {3x - 4{x^3}} \right) is
View answer and explanation
Correct option: D
23\frac{2}{3}
Option D (2/3) is correct because by substituting x=sin⁡θx = \sin \theta, the expression simplifies, and the derivative can be easily found.
850 Medium
850 of 932
Application +4 | -1
If xmyn=(x+y)m+n{x^m}{y^n} = {\left( {x + y} \right)^{m + n}}, then (dy/dx)x=1,y=2{\left( {dy/dx} \right)_{x = 1,y = 2}} is equal to
View answer and explanation
Correct option: B
2
Option B (2) is correct because implicit differentiation and substituting x=1, y=2 leads to dy/dx = 2.
851 Medium
851 of 932
Application +4 | -1
If gg is inverse of ff and f′(x)=11+xnf^{\prime}(x)=\frac{1}{1+x^n}, then g′(x)g^{\prime}(x) equals
View answer and explanation
Correct option: C
1+[g(x)]n1+[g(x)]^n
Option C is correct because it correctly applies the inverse function derivative formula and substitutes the given function's derivative.
852 Medium
852 of 932
Application +4 | -1
If G(x)=125−x2\mathrm{G}(\mathrm{x})=\frac{1}{\sqrt{25-\mathrm{x}^2}}, then xx→0G(x+h)−G(4)x−4x_{x \rightarrow 0} \frac{\mathrm{G}(\mathrm{x}+\mathrm{h})-\mathrm{G}(4)}{\mathrm{x}-4} has the value
View answer and explanation
Correct option: D
none of these
Option D (none of these) is correct because the limit represents the derivative of G(x) at x=4, and the calculated derivative does not match any of the given options.
853 Easy
853 of 932
Application +4 | -1
If f(x)=xnf(x)=x^n, then the value of f(1)+f′(1)1!+f′′(1)2!+f′′′(1)3!+….+f(n)(1)n!f(1)+\frac{f^{\prime}(1)}{1 !}+\frac{f^{\prime \prime}(1)}{2 !}+\frac{f^{\prime \prime \prime}(1)}{3 !}+\ldots .+\frac{f^{(n)}(1)}{n !}, is
View answer and explanation
Correct option: B
2n2^n
Option B is correct because it represents the sum of the binomial coefficients, which equals 2^n.
854 Easy
854 of 932
Analyzing +4 | -1
If f(x)=cos⁡2x+cos⁡2(x+π3)+sin⁡xsin⁡(x+π3)f\left( x \right) = {\cos ^2}x + {\cos ^2}\left( {x + \frac{\pi }{3}} \right) + \sin x\sin \left( {x + \frac{\pi }{3}} \right) and g(54)=3g\left( {\frac{5}{4}} \right) = 3, then ddx(gof(x))=\frac{d}{{dx}}\left( {gof\left( x \right)} \right) =
View answer and explanation
Correct option: B
0
Option B (0) is correct because f(x) simplifies to a constant, 54\frac{5}{4}, making gof(x) also a constant, and the derivative of a constant is 0.
855 Easy
855 of 932
Analyzing +4 | -1
Let f(xy)=f(x)f(y)f(x y)=f(x) f(y) for all x>0,y>0x>0, y>0 and f(1+x)=1+x{1+g(x)}f(1+x)=1+x\{1+g(x)\}, where lim⁡x→0g(x)=0\quad \lim\limits _{x \rightarrow 0} g(x)=0, then ∫f(x)f′(x)dx\int \frac{f(x)}{f^{\prime}(x)} d x, is
View answer and explanation
Correct option: A
x22+C\frac{x^2}{2}+C
Option A (x22+C\frac{x^2}{2}+C) is correct because after finding f(x)=xf(x) = x, the integral ∫f(x)f′(x)dx\int \frac{f(x)}{f^{\prime}(x)} dx simplifies to ∫xdx\int x dx, which equals x22+C\frac{x^2}{2}+C.

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856 Medium
856 of 932
Application +4 | -1
The interval of increase of the function y=y= x−2sin⁡xx-2 \sin x if 0≤x≤2π0 \leq x \leq 2 \pi, is
View answer and explanation
Correct option: D
(π/3,5π/3)(\pi / 3,5 \pi / 3)
Option D (π3,5π3\frac{{\pi}}{{3}}, \frac{{5 \pi}}{{3}}) is correct because the function is increasing where its derivative is positive, which occurs in this interval.
857 Easy
857 of 932
Application +4 | -1
If y=tan⁡−1[sin⁡x+cos⁡xcos⁡x−sin⁡x]y=\tan ^{-1}\left[\frac{\sin x+\cos x}{\cos x-\sin x}\right], then dy/dx=d y / d x=
View answer and explanation
Correct option: D
1
Option D (1) is correct because simplifying the expression inside the arctangent yields tan⁡(π4+x)\tan(\frac{{\pi}}{{4}} + x), so y=π4+xy = \frac{{\pi}}{{4}} + x, and dydx=1\frac{{dy}}{{dx}} = 1.
858 Easy
858 of 932
Application +4 | -1
If x=a(cos⁡t+log⁡tan⁡t2),y=asin⁡tx=a\left(\cos t+\log \tan \frac{t}{2}\right), y=a \sin t, then dydx\frac{d y}{d x} is equal to
View answer and explanation
Correct option: A
tan⁡t\tan t
Option A (tan⁡t\tan t) is correct because it is the simplified form of dydx\frac{dy}{dx} after applying trigonometric identities.
859 Easy
859 of 932
Application +4 | -1
For x∈(0,14),x \in \left( {0,\frac{1}{4}} \right), if the derivative of tan⁡−1(6xx1−9x3){\tan ^{ - 1}}\left( {\frac{{6x\sqrt x }}{{1 - 9{x^3}}}} \right) is x⋅g(x)\sqrt x \cdot g\left( x \right), then g(x)g(x) equals
View answer and explanation
Correct option: A
91+9x3\frac{9}{{1 + 9{x^3}}}
Option D (31+9x3\frac{3}{{1 + 9{x^3}}}) is incorrect. Option A (91+9x3\frac{9}{{1 + 9{x^3}}}) is correct because it is the derivative of the given function after simplification using the arctangent double angle formula.
860 Medium
860 of 932
Application +4 | -1
If variables xx and yare related by the equation x=∫0y11+9u2du,x = \int \nolimits_0^y \frac{1}{{\sqrt {1 + 9{u^2}} }}du, then d2ydx2\frac{{{d^2}y}}{{d{x^2}}} is equal to
View answer and explanation
Correct option: C
9y9y
Option C (9y9y) is correct because after differentiating the given equation twice and simplifying, we find that d2ydx2=9y\frac{{{d^2}y}}{{d{x^2}}} = 9y.
861 Medium
861 of 932
Application +4 | -1
If f(x)=x3+2x2 f′(1)+xf′′(2)+f′′′(3).f(x) = x^3 + 2x^2\,f^\prime(1) + xf^{\prime\prime}(2) + f^{\prime\prime\prime}(3). The value of f′(10)f^\prime(10) is equal to _____.
View answer and explanation
Correct option: A
218
Option A (218) is correct because it is the value of f′(10)f'(10) obtained after finding f′(1)f'(1), f′′(2)f''(2), and substituting them into the expression for f′(x)f'(x).
862 Medium
862 of 932
Application +4 | -1
If y=cos⁡23x2−sin⁡23x2y = {\cos ^2}\frac{{3x}}{2} - {\sin ^2}\frac{{3x}}{2}, then d2ydx2\frac{{{d^2}y}}{{d{x^2}}}
View answer and explanation
Correct option: C
−9y- 9y
Option C (-9y) is correct because the second derivative of y = cos(3x) is -9cos(3x), which is -9y.
863 Medium
863 of 932
Application +4 | -1
If y=sin⁡−1{x−ax−a−ax}y = {\sin ^{ - 1}}\left\{ {\sqrt {x - ax} - \sqrt {a - ax} } \right\}, then dydx\frac{{dy}}{{dx}} is equal to
View answer and explanation
Correct option: C
12x(1−x)\frac{1}{{2\sqrt {x\left( {1 - x} \right)} }}
Option C (12x(1−x)\frac{1}{{2\sqrt {x\left( {1 - x} \right)} }}) is correct because it is the derivative of the simplified inverse sine function after applying trigonometric identities.
864 Medium
864 of 932
Application +4 | -1
ddx[log⁡(tan⁡x)]\frac{d}{d x}[\log (\tan x)] is equal to
View answer and explanation
Correct option: D
2cosec⁡2x2 \operatorname{cosec} 2 x
Option D (2cosec⁡2x2 \operatorname{cosec} 2 x) is correct because differentiating and simplifying the expression leads to this result.
865 Easy
865 of 932
Analyzing +4 | -1
If y=x+x2+x3…∞,y = x + {x^2} + {x^3} \ldots \infty , where ∣x∣<1,\left| x \right| < 1, then for ∣y∣<1,dxdy\left| y \right| < 1,\frac{{dx}}{{dy}} is equal to
View answer and explanation
Correct option: C
1−2y+3y2−...∞1 - 2y + 3{y^2} - ...\infty
The correct answer is C (1−2y+3y2−...∞1 - 2y + 3{y^2} - ...\infty) because it is derived by expressing x in terms of y and then differentiating the resulting series.
866 Easy
866 of 932
Analyzing +4 | -1
If f(x)=11−xf(x)=\frac{1}{1-x}, then the derivative of the composite function f[f{f(x)}]f[f\{f(x)\}] is equal to
View answer and explanation
Correct option: C
1
Option C (1) is correct because the composite function simplifies to f[f{f(x)}] = x, and the derivative of x is 1.
867 Easy
867 of 932
Application +4 | -1
If y=a+bx3/2x5/4y = \frac{{a + b{x^{3/2}}}}{{{x^{5/4}}}} and y′=0y' = 0 at x=5x = 5, then the ratio a:ba:b is equal to
View answer and explanation
Correct option: A
5:1\sqrt 5 :1
Option A (√5 : 1) is correct because setting the derivative to zero at x=5 and solving for the ratio a:b gives √5 : 1.
868 Medium
868 of 932
Analyzing +4 | -1
The derivative of f(x)=∣x∣3f\left( x \right) = {\left| x \right|^3} at x=0x = 0 is
View answer and explanation
Correct option: B
0
Option B (0) is correct because the left-hand derivative and right-hand derivative of the function at x=0 both equal 0, indicating the derivative exists and is 0.
869 Easy
869 of 932
Application +4 | -1
If x=ey+e+⋯xx>0x=e^{y+e+\cdots x} x>0, then dydx\frac{d y}{d x} is
View answer and explanation
Correct option: A
1−xx\frac{1-x}{x}
Option A ((1-x)/x) is correct because it's the derivative dy/dx obtained after logarithmic differentiation.
870 Easy
870 of 932
Analyzing +4 | -1
If y=1−x+x22!−x33!+x44!−…,y = 1 - x + \frac{{{x^2}}}{{2!}} - \frac{{{x^3}}}{{3!}} + \frac{{{x^4}}}{{4!}} - \ldots , then d2ydx2\frac{{{d^2}y}}{{d{x^2}}} is equal to
View answer and explanation
Correct option: C
yy
Option C (yy) is correct because the given series is the Taylor series expansion of e−xe^{-x}, and the second derivative of e−xe^{-x} is e−xe^{-x}, which is equal to yy.
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