Differentiation Techniques - Class XI

Question 1 Multiple Choice (Single Answer)

Differentiate $\tan^{-1} \sqrt{\dfrac{1+\cos x}{1- \cos x}}$

$\dfrac{-1}{2}$
<span>$\dfrac{1}{4}$</span>
<span>$\dfrac{1}{8}$</span>
None of these

Question 2 Multiple Choice (Single Answer)

If $y=\dfrac{1+x^2+x^4}{1+x+x^2}$ and $\dfrac{dy}{dx}=ax+b$, then the values of $a$ and $b$ are,

$a=2,b=1$
$a=-2,b=1$
$a=2,b=-1$
$a=-2,b=-1$

Question 3 Multiple Choice (Single Answer)

If $x=1(\theta+sin,\theta), y=a(1-cos \theta)$, then at $\theta=\dfrac{\pi}{2},y'=\dfrac{2}{a}$.

True
False

Question 4 Multiple Choice (Single Answer)

If $f\left( x \right) =|x-2|,g\left( x \right) =f\left( f\left( x \right) \right) $, then for $x>4$, $g'(x)=$

0
1
-1
2

Question 5 Multiple Choice (Single Answer)

Differential coefficient of $\log\ \sin x$ is :

$\cos x$
$\tan x$
$\text{cosec} ,x$
$\cot x$

Question 6 Multiple Choice (Single Answer)

If $f\left( x \right) =\sqrt { { x }^{ 2 }-2x+1 } $, then

$f^{ ' }\left( x \right) =1,\forall x$
$f^{ ' }\left( x \right) =1, \forall x\ge 1$
$f^{ ' }\left( x \right) =1, \forall x\le 1$
$f^{ ' }\left( x \right) =1,if\quad x>1\quad and\quad f^{ ' }\left( x \right) =-1\quad if\quad x<1$

Question 7 Multiple Choice (Single Answer)

Derivative of $(\sin x)^x + \sin^{-1} \sqrt{x}$ with respect to $x$ is

$(x \cot x + \log \sin x) + \dfrac{1}{2\sqrt{x - x^2}}$
$(x \cot x + \log \sin x) + \dfrac{1}{\sqrt{x - x^2}}$
$(\sin x)^x (x \cot x + \log ,x) + \dfrac{1}{\sqrt{x - x^2}}$
$(\sin x)^x (x \cot x + \log \sin x) + \dfrac{1}{2\sqrt{x - x^2}}$

Question 8 Multiple Choice (Single Answer)

Let f(x) be a differentiable function satisfying $f(x+y)=f(x)+f(y)\forall x, y \in R$ and $f(0)=1$ then $\displaystyle\lim _{x\rightarrow 0}\dfrac{2^{f(\tan^2x)}-2^{f(\sin^2x)}}{x^3f(\sin x)}$ equals to?

$\dfrac{1}{2} ln2$
$ln 2$
$\dfrac{1}{4}ln 2$
$\dfrac{1}{8} ln2$

Question 9 Multiple Choice (Single Answer)

If $t={ \sin { } }^{ -1 }{ 2 }^{ s }$ Then $\dfrac { ds }{ dt }$ is equal to

$\dfrac { \log { 2 } }{ \sqrt { 1-t^{ 2 } } }$
$\dfrac { \sin { t } }{ \log { 2 } }$
$\dfrac { \cot { t } }{ \log { 2 } }$
None of these

Question 10 Multiple Choice (Single Answer)

If $u=e^{x}(xcosy-ysiny)$ then $\frac{d^{2}y}{dx^{2}}+\frac{d^{2}u}{dy^{2}}=0$.

True
False

Question 11 Multiple Choice (Single Answer)

If $U=tan^{-1}(\dfrac{x^3+y^3}{x+y})$ , then $x\dfrac{du}{dx}+y\dfrac{du}{dy}=sinu$.

True
False

Question 12 Multiple Choice (Single Answer)

If $y=\sqrt{x}-\dfrac{1}{\sqrt{x}}$, then $2x\dfrac{dy}{dx}+y$=

$\sqrt{x}$
$2\sqrt{x}$
$3\sqrt{x}$
$\dfrac{\sqrt{x}}{2}$

Question 13 Multiple Choice (Single Answer)

If $x\sqrt {1+y}+y\sqrt {1+x}=0$ then $\dfrac {dy}{dx}=\dfrac {1}{(1+x)^{2}}$

True
False

Question 14 Multiple Choice (Single Answer)

Given $y = x \sqrt{x^2+1}, \dfrac{dy}{dx}$=

$ \sqrt{x^2+1}$
$\dfrac{2x^2+1}{ \sqrt{x^2+1}}$
$\dfrac{3x^2+1}{ \sqrt{x^2+1}}$
$\dfrac{3x^2+2}{ \sqrt{x^2+1}}$

Question 15 Multiple Choice (Single Answer)

If $\sin { { y+e }^{ -x\cos { y } } } =e\quad then\quad \frac { dy }{ dx } \quad at\quad (1,\pi )$ is equal to

$\sin { y } $
$-x\cos { y } $
$e$
$\sin { y } -x\cos { y } $

Question 16 Multiple Choice (Single Answer)

If $\displaystyle \cos^{-1}\left ( \frac{x^{2}-y^{2}}{x^{2}+y^{2}} \right )=\log a$ then $\displaystyle \frac{dy}{dx}$ is equal to

$\dfrac{y}{x}$
$\dfrac{x}{y}$
$\displaystyle\dfrac{ x^{2}}{y^{2}}$
$\displaystyle\dfrac{ y^{2}}{x^{2}}$

Question 17 Multiple Choice (Single Answer)

If $\displaystyle y=\sec(\tan^{-1}x)$, then $\displaystyle \frac {dy}{dx}$ at $x=1$ is equal to

$\displaystyle \frac {1}{\sqrt {2}}$
$\displaystyle \frac {1}{2}$
$1$
$\sqrt {2}$

Question 18 Multiple Choice (Single Answer)

If $y=sec(tan^{-1}x)$, then $\displaystyle\frac{dy}{dx}$ is.

$\displaystyle\frac{x}{\sqrt{1+x^2}}$
$\displaystyle\frac{-x}{\sqrt{1+x^2}}$
$\displaystyle\frac{x}{\sqrt{1-x^2}}$
None of these

Question 19 Multiple Choice (Single Answer)

If ${ x }^{ 2 }.{ e }^{ y }+2x{ ye }^{ x }+13=0$, then $\dfrac { dy }{ dx }$ is

$\dfrac { -2x{ e }^{ y-x }-2y\left( x+1 \right) }{ x\left( { xe }^{ y-x }+2 \right) }$
$\dfrac { 2x{ e }^{ x-y }+2y\left( x+1 \right) }{ x\left( { xe }^{ y-x }+2 \right) }$
$\dfrac { -2x{ e }^{ x-y }+2y\left( x+1 \right) }{ x\left( { xe }^{ y-x }+2 \right) }$
$None\ of\ these$

Question 20 Multiple Choice (Single Answer)

Find: $\dfrac{d}{{\text dx}}\left( {\dfrac{{1 - \cos x}}{{\sin x}}} \right) ,,,$

${\sec ^2}\dfrac{x}{2}$
$,\dfrac{1}{2}{\sec ^2}\dfrac{x}{2},$
$,,2{\sec ^2}\dfrac{x}{2}$
$,,3{\sec ^2}\dfrac{x}{2}$

Question 21 Multiple Choice (Single Answer)

Derivative of ${ \log { x } }^{ \cos { x } }$ with respect to $x$ is

${ \log { x } }^{ \cos { x } }\left[ \cfrac { \cos { x } }{ x\log { x } } -\sin { x } \log { \left( \log { x } \right) } \right] \quad $
${ \log { x } }^{ \cos { x } }\left[ \cfrac { \cos { x } }{ x\log { x } } -\cos { x } \log { \left( \log { x } \right) } \right] \quad $
${ \log { x } }^{ \sin { x } }\left[ \cfrac { \sin { x } }{ x\log { x } } -\sin { x } \log { \left( \log { x } \right) } \right] \quad $
None of these

Question 22 Multiple Choice (Single Answer)

If $xe^{xy}-y=\sin x$, then $\dfrac {dy}{dx}$ at $x=0$ is

$0$
$1$
$-1$
$None\ of\ these$

Question 23 Multiple Choice (Single Answer)

$\dfrac {d}{dx}(x^{\ell n x})$ is equal to

$2x^{\ell n x-1}\ell n x$
$x^{\ell n x-1}$
$2/3(\ell n x)$
$x^{\ell n x-1}.\ell n x$

Question 24 Multiple Choice (Single Answer)

If $y=(tan , x)^{(tan, x)^{tan,x}}, $ then at $x=\dfrac{\pi}{4}, \dfrac{dy}{dx}$ is equal to

0
3
2
None of these

$y={ \left( tanx \right) }^{ { \left( tanx \right) }^{ \left( tanx \right) } }$

We have to find $\dfrac { dy }{ dx } $ at $x=\dfrac { \Pi }{ 4 } $.

Now consider:

$y={ f\left( x \right) }^{ g\left( x \right) }$

$ln\left( y \right) =g\left( x \right) lnf\left( x \right) $

$\Rightarrow \dfrac { 1 }{ y } \dfrac { dy }{ dx } =g\left( x \right) \dfrac { { f }^{ 1 }\left( x \right) }{ f\left( x \right) } +{ g }^{ 1 }\left( x \right) lnf\left( x \right) $

$\Rightarrow \dfrac { dy }{ dx } =y\left[ g\left( x \right) \dfrac { { f }^{ 1 }\left( x \right) }{ f\left( x \right) } +{ g }^{ 1 }\left( x \right) lnf\left( x \right) \right] $

$\Rightarrow \dfrac { dy }{ dx } ={ f\left( x \right) }^{ g\left( x \right) }\left[ { g }^{ 1 }\left( x \right) lnf\left( x \right) +g\left( x \right) \dfrac { { f }^{ 1 }\left( x \right) }{ f\left( x \right) } \right] $

So, We have

$y={ \left( tanx \right) }^{ { \left( tanx \right) }^{ tanx } }$

Where $f\left( x \right) =tan\left( x \right) $

${ f }^{ 1 }\left( x \right) ={ \sec }^{ 2 }x$

$g\left( x \right) ={ \left( tanx \right) }^{ tanx }$

${ g }^{ 1 }\left( x \right) ={ \left( tanx \right) }^{ tanx\left[ { \sec }^{ 2 }xln\left( tan\left( x \right) \right) +\dfrac { \tan\left( x \right) { \sec }^{ 2 }x }{ tanx } \right] }$

Now, $\tan\left( \dfrac { \Pi }{ 4 } \right) =1$ and $\sec\left( \Pi /4 \right) =\sqrt { 2 } $.

$f\left( \Pi /4 \right) =1$

${ f }^{ 1 }\left( \Pi /4 \right) =2$

$g\left( \Pi /4 \right) =1$

${ g }^{ 1 }\left( \Pi /4 \right) =1\left( 2ln\left( 1 \right) +2 \right) =2$

$\dfrac { dy }{ dx } $ at $\Pi /4$ is

$\dfrac { dy }{ dx } ={ f\left( \Pi /4 \right) }^{ g\left( \Pi /4 \right) \left[ { g }^{ 1 }\left( \Pi /4 \right) ln\left( f\left( \Pi /4 \right) \right) +g\left( \Pi /4 \right) \dfrac { { f }^{ 1 }\left( \Pi /4 \right) }{ f\left( \Pi /4 \right) } \right] }$

$\Rightarrow \dfrac { dy }{ dx } =1\left[ 2ln\left( 1 \right) +1\times 2 \right] =2$

$\Rightarrow \dfrac { dy }{ dx } =2$.

Question 25 Multiple Choice (Single Answer)

If $x=a\sin \theta$ and $y=b\cos\theta$, then $\displaystyle\frac{d^2y}{dx^2}$ is

$\displaystyle\frac{a}{b^2}\sec^2\theta$
$\displaystyle\frac{-b}{a}\sec^2\theta$
$\displaystyle\frac{b}{a^2}\sec^3\theta$
$\displaystyle\frac{-b}{a^2}\sec^3\theta$

Differentiation Techniques - Class XI

Questions