Advertisements
Advertisements
प्रश्न
Differentiate \[\sin^{- 1} \left( 2x \sqrt{1 - x^2} \right)\] with respect to \[\sec^{- 1} \left( \frac{1}{\sqrt{1 - x^2}} \right)\], if \[x \in \left( \frac{1}{\sqrt{2}}, 1 \right)\] ?
Advertisements
उत्तर
\[\text { Let, u }= \sin^{- 1} \left( 2x\sqrt{1 - x^2} \right)\]
\[ \text { Put x } = \sin\theta\]
\[ \Rightarrow u = \sin^{- 1} \left( 2\sin\theta\sqrt{1 - \sin^2 \theta} \right)\]
\[ \Rightarrow u = \sin^{- 1} \left( 2 \sin\theta \cos\theta \right) \]
\[ \Rightarrow u = \sin^{- 1} \left( \sin2\theta \right) . . . \left( i \right)\]
\[\text { And, } \]
\[\text { Let, v } = se c^{- 1} \left( \frac{1}{\sqrt{1 - x^2}} \right)\]
\[ \Rightarrow v = se c^{- 1} \left( \frac{1}{\sqrt{1 - \sin^2 \theta}} \right) \]
\[ \Rightarrow v = se c^{- 1} \left( \frac{1}{\cos\theta} \right) \]
\[ \Rightarrow v = se c^{- 1} \left( sec\theta \right) \]
\[ \Rightarrow v = \cos^{- 1} \left( \frac{1}{\frac{1}{\cos\theta}} \right) \left[ \text { Since }, se c^{- 1} x = \cos^{- 1} \left( \frac{1}{x} \right) \right]\]
\[ \Rightarrow v = \cos^{- 1} \left( \cos\theta \right) . . . \left( ii \right)\]
\[\text { Here }, \]
\[ x \in \left( \frac{1}{\sqrt{2}}, 1 \right)\]
\[ \Rightarrow \sin\theta \in \left( \frac{1}{\sqrt{2}}, 1 \right)\]
\[ \Rightarrow \theta \in \left( \frac{\pi}{4}, \frac{\pi}{2} \right)\]
\[\text { So, from equation } \left( i \right), \]
\[ u = 2\theta ........\left[ \text { Since }, \sin^{- 1} \left( \sin\theta \right) = \theta, \text{ if }\theta \in \left( - \frac{\pi}{2}, \frac{\pi}{2} \right) \right] \]
\[\text { Let, u }= 2 \sin^{- 1} x .........\left[ \text { Since,} x = \sin\theta \right]\]
Differentiating it with respect to x,
\[\frac{du}{dx} = 2\left( \frac{1}{\sqrt{1 - x^2}} \right)\]
\[ \Rightarrow \frac{du}{dx} = \frac{2}{\sqrt{1 - x^2}} . . . \left( iii \right)\]
\[\text { And, from equation } \left( ii \right), \]
\[v = \theta \left[ \text{ Since,} \cos^{- 1} \left( \cos\theta \right) = \theta, \text { if } \theta \in \left[ 0, \pi \right] \right]\]
\[ \Rightarrow v = \sin^{- 1} x \left[ \text { Since }, x = \sin\theta \right]\]
Differentiating it with respect to x,
\[\frac{dv}{dx} = \frac{1}{\sqrt{1 - x^2}} . . . \left( iv \right)\]
\[\text {dividing equation } \left( iii \right) by \left( iv \right), \]
\[\frac{\frac{du}{dx}}{\frac{dv}{dx}} = \frac{2}{\sqrt{1 - x^2}} \times \frac{\sqrt{1 - x^2}}{1}\]
\[ \therefore \frac{du}{dv} = 2\]
APPEARS IN
संबंधित प्रश्न
Prove that `y=(4sintheta)/(2+costheta)-theta `
Differentiate the following functions from first principles x2ex ?
Differentiate sin (log x) ?
Differentiate tan 5x° ?
Differentiate \[\sqrt{\frac{a^2 - x^2}{a^2 + x^2}}\] ?
Differentiate \[\sqrt{\frac{1 + x}{1 - x}}\] ?
Differentiate \[\frac{e^{2x} + e^{- 2x}}{e^{2x} - e^{- 2x}}\] ?
Differentiate \[\log \left( \tan^{- 1} x \right)\]?
Differentiate \[\cos \left( \log x \right)^2\] ?
Differentiate \[\tan^{- 1} \left( \frac{a + bx}{b - ax} \right)\] ?
Find \[\frac{dy}{dx}\] in the following case \[x^{2/3} + y^{2/3} = a^{2/3}\] ?
If \[xy \log \left( x + y \right) = 1\] ,Prove that \[\frac{dy}{dx} = - \frac{y \left( x^2 y + x + y \right)}{x \left( x y^2 + x + y \right)}\] ?
If \[y = x \sin \left( a + y \right)\] ,Prove that \[\frac{dy}{dx} = \frac{\sin^2 \left( a + y \right)}{\sin \left( a + y \right) - y \cos \left( a + y \right)}\] ?
Differentiate \[x^{\sin x}\] ?
Differentiate \[x^{\sin^{- 1} x}\] ?
find \[\frac{dy}{dx}\] \[y = \frac{\left( x^2 - 1 \right)^3 \left( 2x - 1 \right)}{\sqrt{\left( x - 3 \right) \left( 4x - 1 \right)}}\] ?
If \[x^{13} y^7 = \left( x + y \right)^{20}\] prove that \[\frac{dy}{dx} = \frac{y}{x}\] ?
If \[x^x + y^x = 1\], prove that \[\frac{dy}{dx} = - \left\{ \frac{x^x \left( 1 + \log x \right) + y^x \cdot \log y}{x \cdot y^\left( x - 1 \right)} \right\}\] ?
If \[e^y = y^x ,\] prove that\[\frac{dy}{dx} = \frac{\left( \log y \right)^2}{\log y - 1}\] ?
If \[y = \sqrt{\cos x + \sqrt{\cos x + \sqrt{\cos x + . . . to \infty}}}\] , prove that \[\frac{dy}{dx} = \frac{\sin x}{1 - 2 y}\] ?
If \[x = 2 \cos \theta - \cos 2 \theta \text{ and y} = 2 \sin \theta - \sin 2 \theta\], prove that \[\frac{dy}{dx} = \tan \left( \frac{3 \theta}{2} \right)\] ?
If \[x = e^{\cos 2 t} \text{ and y }= e^{\sin 2 t} ,\] prove that \[\frac{dy}{dx} = - \frac{y \log x}{x \log y}\] ?
Differentiate (log x)x with respect to log x ?
If \[y = x^x , \text{ find } \frac{dy}{dx} \text{ at } x = e\] ?
If \[y = \tan^{- 1} \left( \frac{1 - x}{1 + x} \right), \text{ find} \frac{dy}{dx}\] ?
If \[\sin^{- 1} \left( \frac{x^2 - y^2}{x^2 + y^2} \right) = \text { log a then } \frac{dy}{dx}\] is equal to _____________ .
Find the second order derivatives of the following function tan−1 x ?
Find the second order derivatives of the following function log (log x) ?
If y = ex cos x, prove that \[\frac{d^2 y}{d x^2} = 2 e^x \cos \left( x + \frac{\pi}{2} \right)\] ?
If y = sin (sin x), prove that \[\frac{d^2 y}{d x^2} + \tan x \cdot \frac{dy}{dx} + y \cos^2 x = 0\] ?
If y = (tan−1 x)2, then prove that (1 + x2)2 y2 + 2x(1 + x2)y1 = 2 ?
If y = ex (sin x + cos x) prove that \[\frac{d^2 y}{d x^2} - 2\frac{dy}{dx} + 2y = 0\] ?
If y = sin (log x), prove that \[x^2 \frac{d^2 y}{d x^2} + x\frac{dy}{dx} + y = 0\] ?
If x = a cos nt − b sin nt and \[\frac{d^2 x}{dt} = \lambda x\] then find the value of λ ?
If y = x + ex, find \[\frac{d^2 x}{d y^2}\] ?
If y = a sin mx + b cos mx, then \[\frac{d^2 y}{d x^2}\] is equal to
Find the minimum value of (ax + by), where xy = c2.
