Advertisements
Advertisements
प्रश्न
The derivative of the function \[\cot^{- 1} \left| \left( \cos 2 x \right)^{1/2} \right| \text{ at } x = \pi/6 \text{ is }\] ______ .
पर्याय
(2/3)1/2
(1/3)1/2
31/2
61/2
उत्तर
(2/3)1/2
\[\text{ We have, y } = \cot^{- 1} \left( \sqrt{\cos 2x} \right)\]
\[\Rightarrow \frac{dy}{dx} = \frac{- 1}{1 + \cos 2x}\frac{d}{dx}\sqrt{\cos 2x}\]
\[ \Rightarrow \frac{dy}{dx} = \frac{- 1}{2 \cos^2 x} \times \frac{1}{2\sqrt{\cos 2x}}\frac{d}{dx}\left( \cos 2x \right)\]
\[ \Rightarrow \frac{dy}{dx} = \frac{- 1}{2 \cos^2 x} \times \frac{1}{2\sqrt{\cos 2x}} \times - 2\sin 2x\]
\[ \Rightarrow \frac{dy}{dx} = \frac{\sin2x}{\cos^2 x \times 2\sqrt{\cos2x}}\]
\[ \Rightarrow \frac{dy}{dx} = \frac{2 \sin x \cos x}{\cos^2 x \times 2\sqrt{\cos2x}}\]
\[ \Rightarrow \frac{dy}{dx} = \frac{\tan x}{\sqrt{\cos2x}}\]
\[\text {So, at x } = \frac{\pi}{6},\text{ we get }\]
\[ \left( \frac{dy}{dx} \right)_{x = \frac{\pi}{6}} = \frac{\tan\left( \frac{\pi}{6} \right)}{\sqrt{\cos2\left( \frac{\pi}{6} \right)}} = \frac{\left( \frac{1}{\sqrt{3}} \right)}{\sqrt{\frac{1}{2}}} = \left( \frac{2}{3} \right)^\frac{1}{2}\]
APPEARS IN
संबंधित प्रश्न
If the sum of the lengths of the hypotenuse and a side of a right triangle is given, show that the area of the triangle is maximum, when the angle between them is 60º.
Differentiate the following functions from first principles log cosec x ?
Differentiate tan2 x ?
Differentiate \[\sin \left( \frac{1 + x^2}{1 - x^2} \right)\] ?
Differentiate \[e^{3 x} \cos 2x\] ?
Differentiate \[\left( \sin^{- 1} x^4 \right)^4\] ?
Differentiate \[e^{ax} \sec x \tan 2x\] ?
Differentiate \[\sin^{- 1} \left\{ \sqrt{\frac{1 - x}{2}} \right\}, 0 < x < 1\] ?
Differentiate \[\sin^{- 1} \left( 1 - 2 x^2 \right), 0 < x < 1\] ?
Differentiate \[\tan^{- 1} \left\{ \frac{x}{a + \sqrt{a^2 - x^2}} \right\}, - a < x < a\] ?
If \[y = \tan^{- 1} \left( \frac{\sqrt{1 + x} - \sqrt{1 - x}}{\sqrt{1 + x} + \sqrt{1 - x}} \right), \text{find } \frac{dy}{dx}\] ?
Find \[\frac{dy}{dx}\] in the following case \[e^{x - y} = \log \left( \frac{x}{y} \right)\] ?
If \[\sqrt{1 - x^2} + \sqrt{1 - y^2} = a \left( x - y \right)\] , prove that \[\frac{dy}{dx} = \frac{\sqrt{1 - y^2}}{1 - x^2}\] ?
If \[xy = 1\] prove that \[\frac{dy}{dx} + y^2 = 0\] ?
If \[\tan^{- 1} \left( \frac{x^2 - y^2}{x^2 + y^2} \right) = a\] Prove that \[\frac{dy}{dx} = \frac{x}{y}\frac{\left( 1 - \tan a \right)}{\left( 1 + \tan a \right)}\] ?
Differentiate \[\left( \tan x \right)^{1/x}\] ?
Differentiate\[\left( x + \frac{1}{x} \right)^x + x^\left( 1 + \frac{1}{x} \right)\] ?
Differentiate \[x^{x^2 - 3} + \left( x - 3 \right)^{x^2}\] ?
Find \[\frac{dy}{dx}\] \[y = e^x + {10}^x + x^x\] ?
If \[x^{13} y^7 = \left( x + y \right)^{20}\] prove that \[\frac{dy}{dx} = \frac{y}{x}\] ?
If \[x^m y^n = 1\] , prove that \[\frac{dy}{dx} = - \frac{my}{nx}\] ?
Find \[\frac{dy}{dx}\] ,when \[x = \frac{e^t + e^{- t}}{2} \text{ and } y = \frac{e^t - e^{- t}}{2}\] ?
If \[x = a\sin2t\left( 1 + \cos2t \right) \text { and y } = b\cos2t\left( 1 - \cos2t \right)\] , show that at \[t = \frac{\pi}{4}, \frac{dy}{dx} = \frac{b}{a}\] ?
Differentiate log (1 + x2) with respect to tan−1 x ?
If \[y = \sin^{- 1} \left( \sin x \right), - \frac{\pi}{2} \leq x \leq \frac{\pi}{2}\] ,Then, write the value of \[\frac{dy}{dx} \text{ for } x \in \left( - \frac{\pi}{2}, \frac{\pi}{2} \right) \] ?
If \[y = \log_a x, \text{ find } \frac{dy}{dx} \] ?
If \[y = \log \sqrt{\tan x}, \text{ write } \frac{dy}{dx} \] ?
If \[y = \log \sqrt{\tan x}\] then the value of \[\frac{dy}{dx}\text { at }x = \frac{\pi}{4}\] is given by __________ .
If \[y = \log \left( \frac{1 - x^2}{1 + x^2} \right), \text { then } \frac{dy}{dx} =\] __________ .
If y = tan−1 x, show that \[\left( 1 + x^2 \right) \frac{d^2 y}{d x^2} + 2x\frac{dy}{dx} = 0\] ?
Find \[\frac{d^2 y}{d x^2}\] where \[y = \log \left( \frac{x^2}{e^2} \right)\] ?
If y = ae2x + be−x, show that, \[\frac{d^2 y}{d x^2} - \frac{dy}{dx} - 2y = 0\] ?
\[\text { If x } = a\left( \cos2t + 2t \sin2t \right)\text { and y } = a\left( \sin2t - 2t \cos2t \right), \text { then find } \frac{d^2 y}{d x^2} \] ?
If x = 2at, y = at2, where a is a constant, then find \[\frac{d^2 y}{d x^2} \text { at }x = \frac{1}{2}\] ?
If y = |x − x2|, then find \[\frac{d^2 y}{d x^2}\] ?
If y = sin (m sin−1 x), then (1 − x2) y2 − xy1 is equal to
If y = etan x, then (cos2 x)y2 =
If \[\frac{d}{dx}\left[ x^n - a_1 x^{n - 1} + a_2 x^{n - 2} + . . . + \left( - 1 \right)^n a_n \right] e^x = x^n e^x\] then the value of ar, 0 < r ≤ n, is equal to
Differentiate sin(log sin x) ?
Differentiate the following with respect to x:
\[\cot^{- 1} \left( \frac{1 - x}{1 + x} \right)\]
