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JEE • MATHEMATICS • DIFFERENTIATION

Differentiation practice · Mixed difficulty

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Showing 181-210 of 932 questions

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181 Easy
181 of 932
Application +4 | -1
ddxlog⁡(log⁡x)\frac{d}{d x} \log (\log x) is equal to
View answer and explanation
Correct option: C
(xlog⁡x)−1(x \log x)^{-1}
Option C ((xlog⁡x)−1(x \log x)^{-1}) is correct because the derivative of log⁡(log⁡x)\log(\log x) is 1log⁡x⋅1x\frac{1}{\log x} \cdot \frac{1}{x}, which simplifies to (xlog⁡x)−1(x \log x)^{-1}.
182 Medium
182 of 932
Application +4 | -1
If x+y+y−x=c\sqrt{x+y}+\sqrt{y-x}=c, then d2ydx2\frac{d^2 y}{d x^2} is equal to
View answer and explanation
Correct option: C
−2c2-\frac{2}{c^2}
Option C (2c2\frac{2}{c^2}) is correct because differentiating the given equation twice leads to this result.
183 Easy
183 of 932
Application +4 | -1
If y=sin⁡−1(sin⁡αsin⁡x1−cos⁡αsin⁡x)y=\sin ^{-1}\left(\frac{\sin \alpha \sin x}{1-\cos \alpha \sin x}\right), then y′(0)y^{\prime}(0) is
View answer and explanation
Correct option: D
sin⁡α\sin \alpha.
Option D (sin⁡α\sin \alpha) is correct because it is the value of the derivative of yy with respect to xx evaluated at x=0x=0.
184 Easy
184 of 932
Application +4 | -1
Let f(x)=118−x2f\left( x \right) = \frac{1}{{\sqrt {18 - {x^2}} }}. The value of lim⁡x→3f(x)−f(3)x−3\lim\limits_{x \to 3} \frac{{f\left( x \right) - f\left( 3 \right)}}{{x - 3}}, is
View answer and explanation
Correct option: D
19\frac{1}{9}
Option D (19\frac{1}{9}) is correct because the limit is the derivative of f(x)f(x) evaluated at x=3x=3, which equals 19\frac{1}{9}.
185 Medium
185 of 932
Application +4 | -1
If y=sin⁡−1(sin⁡αsin⁡x1−cos⁡αsin⁡x)y = {\sin ^{ - 1}}\left( {\frac{{\sin \alpha \sin x}}{{1 - \cos \alpha \sin x}}} \right), then y′(0)y'\left( 0 \right) is
View answer and explanation
Correct option: D
sin⁡α\sin \alpha
Option D (sin⁡α\sin \alpha) is the correct derivative at x=0, obtained by differentiating the given function and substituting x=0.
186 Medium
186 of 932
Application +4 | -1
If ey+ey+…∞,x>0{e^{y + {e^{y + \ldots \infty }}}},x > 0 then dydx\frac{{dy}}{{dx}} is
View answer and explanation
Correct option: C
1−xx\frac{{1 - x}}{x}
Option C (1−xx\frac{{1 - x}}{x}) is correct because it is the derivative obtained by implicit differentiation of the given equation.
187 Medium
187 of 932
Application +4 | -1
If y=(sin⁡−1x)2+(cos⁡−1x)2\mathrm{y}=\left(\sin ^{-1} \mathrm{x}\right)^2+\left(\cos ^{-1} \mathrm{x}\right)^2, then (1−x2)y2−xy1=\left(1-\mathrm{x}^2\right) \mathrm{y_2-x y_1}=
View answer and explanation
Correct option: B
4
Option B (4) is correct because, after differentiating the given equation twice and simplifying, the expression (1-x2)y_2 - xy_1 equals 4.
188 Medium
188 of 932
Application +4 | -1
If xy=ax2+(b/x)x y=a x^2+(b / x), then xd2ydx2+2dydx=x \frac{d^2 y}{d x^2}+2 \frac{d y}{d x}=
View answer and explanation
Correct option: C
2yx\frac{2 y}{x}
Option C (2yx\frac{{2 y}}{{x}}) is correct because after differentiating the given equation twice and simplifying, the expression xd2ydx2+2dydxx \frac{{d^2 y}}{{d x^2}}+2 \frac{{d y}}{{d x}} simplifies to 2yx\frac{{2 y}}{{x}}.
189 Easy
189 of 932
Application +4 | -1
If f(x)=1+cos⁡2(x2),f\left( x \right) = \sqrt {1 + {{\cos }^2}\left( {{x^2}} \right)} , then f′(π2)f'\left( {\frac{{\sqrt \pi }}{2}} \right) is
View answer and explanation
Correct option: B
−π6- \sqrt {\frac{\pi }{6}}
Option B (−π6- \sqrt {\frac{\pi }{6}}) is correct because it's the result of applying the chain rule and evaluating the derivative at the given point.
190 Medium
190 of 932
Analyzing +4 | -1
Let S be the set of all points in (−π,π)(-\pi, \pi) at which the function f(x)=min⁡∣sin⁡x,cos⁡x∣f(x)=\min |\sin x, \cos x| is not differentiable. Then SS is a subset of which of the following?
View answer and explanation
Correct option: A
{−3π4,−π4,π4,3π4}\left\{\frac{-3 \pi}{4}, \frac{-\pi}{4}, \frac{\pi}{4}, \frac{3 \pi}{4}\right\}
Option A is correct because the function is not differentiable at points where the absolute values of sine and cosine intersect within the given interval, which are −3π4,−π4,π4,3π4\frac{-3 \pi}{4}, \frac{-\pi}{4}, \frac{\pi}{4}, \frac{3 \pi}{4}.
191 Easy
191 of 932
Application +4 | -1
If a function is represented parametrically by the equations
x=1+tt3,y=32t2+2tx=\frac{1+t}{t^3}, \quad y=\frac{3}{2 t^2}+\frac{2}{t} then
View answer and explanation
Correct option: B
x(dydx)3=1+dydxx\left(\frac{d y}{d x}\right)^3=1+\frac{d y}{d x}
Option B (x(dydx)3=1+dydxx(\frac{{d y}}{{d x}})^3=1+\frac{{d y}}{{d x}}) is correct because after finding dydx\frac{{dy}}{{dx}} and substituting in the given equation, it satisfies the equation.
192 Medium
192 of 932
Understanding +4 | -1
Let g(x)g\left( x \right) be the inverse of an invertible function f(x)f\left( x \right) which is differentiable at x=cx = c, then g′(f(c))g'\left( {f\left( c \right)} \right) equals
View answer and explanation
Correct option: B
1f′(c)\frac{1}{{f'\left( c \right)}}
Option B (1f′(c)\frac{1}{{f'(c)}}) is correct because it is the direct application of the inverse function derivative rule.
193 Easy
193 of 932
Application +4 | -1
If 2x+2y=2x+y2^x+2^y=2^{x+y}, then the value of dydx\frac{d y}{d x} at x=yx=y =1=1 is
View answer and explanation
Correct option: B
-1
Option B (-1) is correct because after differentiating the given equation and substituting x=y=1, the value of dy/dx is -1.
194 Medium
194 of 932
Application +4 | -1
If xy⋅yx=16x^y \cdot y^x=16, then dydx\frac{d y}{d x} at (2,2)(2,2) is
View answer and explanation
Correct option: A
-1
Option A (-1) is correct because it is the value of dydx\frac{{dy}}{{dx}} at the point (2,2) after differentiating the given equation implicitly.
195 Medium
195 of 932
Application +4 | -1
If y=sin⁡−1(x1−x+x1−x2)y=\sin ^{-1}\left(x \sqrt{1-x}+\sqrt{x} \sqrt{1-x^{2}}\right), then dydx\frac{d y}{d x} is equal to
View answer and explanation
Correct option: C
11−x2+12x−x2\frac{1}{\sqrt{1-x^{2}}}+\frac{1}{2 \sqrt{x-x^{2}}}
The correct answer is C (11−x2+12x−x2\frac{1}{\sqrt{1-x^{2}}}+\frac{1}{2 \sqrt{x-x^{2}}}) because it is the derivative of the simplified form of the given function after using trigonometric substitution.

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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
196 Easy
196 of 932
Analyzing +4 | -1
If y=x+y+x+y+…∞y=\sqrt{x+\sqrt{y+\sqrt{x+\sqrt{y+\ldots \infty}}}}, then dydx\frac{d y}{d x} is equal to
View answer and explanation
Correct option: D
None of these
The correct answer is D (None of these) because the derivative calculated does not match any of the given options.
197 Medium
197 of 932
Application +4 | -1
If x2+y2=1x^2+y^2=1, then
View answer and explanation
Correct option: B
yy′′+(y′)2+1=0y y^{\prime \prime}+\left(y^{\prime}\right)^2+1=0
Option B is correct because it is the result of differentiating the given implicit equation twice and rearranging the terms.
198 Easy
198 of 932
Analyzing +4 | -1
Let f(x)=log⁡e(sin⁡x),(0<x<π)f(x) = {\log _e}(\sin x),(0 < x < \pi ) and g(x)=sin⁡−1(e−x),(x≥0)g(x) = {\sin ^{ - 1}}({e^{ - x}}),(x \ge 0). If α\alpha is positive real number
such that a=(fog)′(α)a = (fog)'(\alpha ) and b=(fog)(α)b = (fog)(\alpha ), then
View answer and explanation
Correct option: B
aα2−bα−a=1a{\alpha ^2} - b\alpha - a = 1
Option B (aα2−bα−a=1a{\alpha ^2} - b\alpha - a = 1) is correct because the given functions are composed and simplified, then differentiated and evaluated at α\alpha to satisfy the equation.
199 Medium
199 of 932
Understanding +4 | -1
If f(x)=(1+x)nf(x)=(1+x)^{n }, then f(0)+f(0)+12f′(0)+…+1n!fn(0)=……f(0)+f(0)+\frac{1}{2} f^{\prime}(0)+\ldots+\frac{1}{n !} f^n(0)=\ldots \ldots
View answer and explanation
Correct option: B
2n2^n
Option B (2n2^n) is correct because the given expression is the binomial expansion of (1+1)n(1+1)^n.
200 Medium
200 of 932
Application +4 | -1
If y=cos⁡−1(3cos⁡x+4sin⁡x5)y=\cos ^{-1}\left(\frac{3 \cos x+4 \sin x}{5}\right), then dydx\frac{d y}{d x}
View answer and explanation
Correct option: B
1
Option B is correct because it simplifies to 1 after applying trigonometric identities and inverse cosine properties.
201 Easy
201 of 932
Application +4 | -1
ddx(sin⁡x1+cos⁡x)=\frac{d}{d x}\left(\frac{\sin x}{1+\cos x}\right)=
View answer and explanation
Correct option: A
12sec⁡2x2\frac{1}{2} \sec ^{2} \frac{x}{2}
Option A (1/2 sec2(x/2)) is correct because the derivative of sin(x)/(1+cos(x)) simplifies to tan(x/2), and the derivative of tan(x/2) is 1/2 sec2(x/2).
202 Easy
202 of 932
Analyzing +4 | -1
The derivative of tan⁡−1(1+x2−1x){\tan ^{ - 1}}\left( {\frac{{\sqrt {1 + {x^2}} - 1}}{x}} \right) with respect to tan⁡−1(2x1−x21−2x2){\tan ^{ - 1}}\left( {\frac{{2x\sqrt {1 - {x^2}} }}{{1 - 2{x^2}}}} \right) at x=12x = \frac{1}{2} is
View answer and explanation
Correct option: D
310 \frac{{\sqrt 3 }}{{10}}\,
Option D is correct because after simplification using trigonometric substitutions and differentiation, the derivative evaluates to 310\frac{{\sqrt 3 }}{{{{10}}}} at x=12x = \frac{1}{2}.
203 Medium
203 of 932
Application +4 | -1
If y=log⁡cos⁡xsin⁡x,y = {\log _{\cos x}}\sin x, then dydx\frac{{dy}}{{dx}} is equal to
View answer and explanation
Correct option: A
(cot⁡xlog⁡cos⁡x+tan⁡xlog⁡sin⁡x)(log⁡cos⁡x)2\frac{{(\cot x\log \cos x + \tan x\log \sin x)}}{{{{(\log \cos x)}^2}}}
Option A is correct because it correctly applies the quotient rule and change of base formula for logarithms.
204 Medium
204 of 932
Analyzing +4 | -1
If f(x)=x(x+x+1)f\left( x \right) = x\left( {\sqrt x + \sqrt {x + 1} } \right), then
View answer and explanation
Correct option: C
f(x)f\left( x \right) is not differentiable at x=0x = 0
Option C is correct because the function is not continuous at x=0 due to the square root of x, and therefore not differentiable.
205 Medium
205 of 932
Application +4 | -1
If x=exp⁡{tan⁡−1(y−x2x2)}x = \exp \left\{ {{{\tan }^{ - 1}}\left( {\frac{{y - {x^2}}}{{{x^2}}}} \right)} \right\}, then dydx\frac{{dy}}{{dx}} equals
View answer and explanation
Correct option: A
2x[1+tan⁡(log⁡x)]+xsec⁡2(log⁡x)2x[1 + \tan (\log x)] + x{\sec ^2}(\log x)
Option A (2x[1+tan⁡(log⁡x)]+xsec⁡2(log⁡x)2x[1 + \tan (\log x)] + x{\sec ^2}(\log x)) is correct because it is the result of differentiating the given equation and solving for dy/dx.
206 Easy
206 of 932
Application +4 | -1
If f(x)=log⁡x{ln⁡(x)}f(x)=\log _x\{\ln (x)\}, then f′(x)f^{\prime}(x) at x=ex=e, is
View answer and explanation
Correct option: D
e−1e^{-1}
Option D (e−1e^{-1}) is correct because after applying the change of base formula and differentiating, the value of the derivative at x=e is 1/e.
207 Medium
207 of 932
Application +4 | -1
If y=(1+x)1/xy=(1+x)^{1 / x}, then dydx=\frac{d y}{d x}=
View answer and explanation
Correct option: B
(1+x)1/x[x1+x−log⁡(1+x)]x2(1+x)^{1 / x} \frac{\left[\frac{x}{1+x}-\log (1+x)\right]}{x^2}
Option B ((1+x)1/x[x1+x−log⁡(1+x)]x2(1+x)^{1 / x} \frac{\left[\frac{x}{1+x}-\log (1+x)\right]}{x^2}) is correct because after applying logarithmic differentiation and simplifying, the derivative is found to be (1+x)1/x[x1+x−log⁡(1+x)]x2(1+x)^{1 / x} \frac{\left[\frac{x}{1+x}-\log (1+x)\right]}{x^2}.
208 Medium
208 of 932
Application +4 | -1
Let f(x)=2/(x+1)f(x)=2 /(x+1) and g(x)=3xg(x)=3 x. It is given that (f∘g)(x0)=(g∘f)(x0)(f \circ g)\left(x_0\right)=(g \circ f)\left(x_0\right). Then (g∘f)′(x0)(g \circ f)^{\prime}\left(x_0\right) equals
View answer and explanation
Correct option: D
−323\frac{-32}{3}
Option D (−323\frac{-32}{3}) is correct because, after finding x0=−14x_0 = -\frac{1}{4}, we use the chain rule to compute (g∘f)′(x0)=g′(f(x0))∗f′(x0)(g \circ f)'(x_0) = g'(f(x_0)) * f'(x_0), which results in −323\frac{-32}{3}.
209 Easy
209 of 932
Application +4 | -1
If f(x)=cot⁡−1(xx−x−x2)f(x)=\cot ^{-1}\left(\frac{x^x-x^{-x}}{2}\right), then f′(1)f^{\prime}(1) equals
View answer and explanation
Correct option: A
−1-1
Option A (-1) is correct because it is the value of the derivative of the given function, evaluated at x=1.
210 Medium
210 of 932
Application +4 | -1
If u=sin⁡−1(x2+y2x+y),u = {\sin ^{ - 1}}\left( {\frac{{{x^2} + {y^2}}}{{x + y}}} \right), then x∂u∂x+y∂u∂yx\frac{{\partial u}}{{\partial x}} + y\frac{{\partial u}}{{\partial y}} is equal to
View answer and explanation
Correct option: B
tan⁡u\tan u
Option B (tan⁡u\tan u) is correct because Euler's theorem is applied to the homogeneous function f(x,y)=sin⁡uf(x,y) = \sin u to find the value of x∂u∂x+y∂u∂yx\frac{{\partial u}}{{\partial x}} + y\frac{{\partial u}}{{\partial y}}.
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