Advertisements
Advertisements
प्रश्न
Differentiate \[\sin^{- 1} \left( \frac{2x}{1 + x^2} \right)\] with respect to \[\tan^{- 1} \left( \frac{2 x}{1 - x^2} \right), \text{ if } - 1 < x < 1\] ?
Advertisements
उत्तर
\[\text { Let, u }= \sin^{- 1} \left( \frac{2x}{1 + x^2} \right)\]
\[\text { Put x } = \tan\theta \Rightarrow \theta = \tan^{- 1} x, \]
\[ \Rightarrow u = \sin^{- 1} \left( \frac{2\tan\theta}{1 + \tan^2 \theta} \right)\]
\[ \Rightarrow u = \sin^{- 1} \left( \sin2\theta \right) . . . \left( i \right)\]
\[\text { Let, v } = \tan^{- 1} \left( \frac{2x}{1 - x^2} \right)\]
\[ \Rightarrow v = \tan^{- 1} \left( \frac{2\tan\theta}{1 - \tan^2 \theta} \right)\]
\[ \Rightarrow v = \tan^{- 1} \left( \tan2\theta \right) . . . \left( ii \right)\]
\[\text { Here, }- 1 < x < 1\]
\[ \Rightarrow - 1 < \tan\theta < 1\]
\[ \Rightarrow - \frac{\pi}{4} < \tan\theta < \frac{\pi}{4}\]
\[\text{ So, from equation } \left( i \right), \]
\[u = 2\theta \left[ \text {Since,} \sin^{- 1} \left( \sin\theta \right) = \theta, \text { if } \theta \in \left[ - \frac{\pi}{2}, \frac{\pi}{2} \right] \right]\]
\[ \Rightarrow u = 2 \tan^{- 1} x\]
Differentiating it with respect to x,
\[\frac{du}{dx} = \frac{2}{1 + x^2} . . . \left( iii \right)\]
\[\text { from equation } \left( ii \right), \]
\[v = 2\theta \left[ \text {Since}, \tan^{- 1} \left( \tan\theta \right) = \theta , \text { if } \theta \in \left( - \frac{\pi}{2}, \frac{\pi}{2} \right) \right]\]
\[ \Rightarrow v = 2 \tan^{- 1} x\]
Differentiating it with respect to x,
\[\frac{dv}{dx} = \frac{2}{1 + x^2} . . . \left( iv \right)\]
\[\text { Dividing equation } \left( iii \right) \text { by } \left( iv \right), \]
\[\frac{\frac{du}{dx}}{\frac{dv}{dx}} = \frac{2}{1 + x^2} \times \frac{1 + x^2}{2}\]
\[ \therefore \frac{du}{dv} = 1\]
APPEARS IN
संबंधित प्रश्न
Differentiate the following functions from first principles x2ex ?
Differentiate \[\log \left( x + \sqrt{x^2 + 1} \right)\] ?
Differentiate \[\frac{2^x \cos x}{\left( x^2 + 3 \right)^2}\]?
Differentiate \[3 e^{- 3x} \log \left( 1 + x \right)\] ?
If \[y = \log \left\{ \sqrt{x - 1} - \sqrt{x + 1} \right\}\] ,show that \[\frac{dy}{dx} = \frac{- 1}{2\sqrt{x^2 - 1}}\] ?
If \[y = \frac{x}{x + 2}\] , prove tha \[x\frac{dy}{dx} = \left( 1 - y \right) y\] ?
Differentiate \[\sin^{- 1} \left\{ \sqrt{1 - x^2} \right\}, 0 < x < 1\] ?
Differentiate \[\tan^{- 1} \left\{ \frac{x}{\sqrt{a^2 - x^2}} \right\}, - a < x < a\] ?
Differentiate \[\sin^{- 1} \left( 2 x^2 - 1 \right), 0 < x < 1\] ?
Differentiate \[\tan^{- 1} \left( \frac{\sqrt{x} + \sqrt{a}}{1 - \sqrt{xa}} \right)\] ?
If \[y = \cot^{- 1} \left\{ \frac{\sqrt{1 + \sin x} + \sqrt{1 - \sin x}}{\sqrt{1 + \sin x} - \sqrt{1 - \sin x}} \right\}\], show that \[\frac{dy}{dx}\] is independent of x. ?
Find \[\frac{dy}{dx}\] in the following case \[\tan^{- 1} \left( x^2 + y^2 \right) = a\] ?
If `ysqrt(1-x^2) + xsqrt(1-y^2) = 1` prove that `dy/dx = -sqrt((1-y^2)/(1-x^2))`
If \[\cos y = x \cos \left( a + y \right), \text{ with } \cos a \neq \pm 1, \text{ prove that } \frac{dy}{dx} = \frac{\cos^2 \left( a + y \right)}{\sin a}\] ?
If \[\sin^2 y + \cos xy = k,\] find \[\frac{dy}{dx}\] at \[x = 1 , \] \[y = \frac{\pi}{4} .\]
Find \[\frac{dy}{dx}\] \[y = e^x + {10}^x + x^x\] ?
Find \[\frac{dy}{dx}\] \[y = \left( \tan x \right)^{\cot x} + \left( \cot x \right)^{\tan x}\] ?
If \[x^{16} y^9 = \left( x^2 + y \right)^{17}\] ,prove that \[x\frac{dy}{dx} = 2 y\] ?
If \[y^x = e^{y - x}\] ,prove that \[\frac{dy}{dx} = \frac{\left( 1 + \log y \right)^2}{\log y}\] ?
If \[y = x \sin \left( a + y \right)\] , prove that \[\frac{dy}{dx} = \frac{\sin^2 \left( a + y \right)}{\sin \left( a + y \right) - y \cos \left( a + y \right)}\] ?
Find the derivative of the function f (x) given by \[f\left( x \right) = \left( 1 + x \right) \left( 1 + x^2 \right) \left( 1 + x^4 \right) \left( 1 + x^8 \right)\] and hence find `f' (1)` ?
If \[y = \sqrt{\log x + \sqrt{\log x + \sqrt{\log x + ... to \infty}}}\], prove that \[\left( 2 y - 1 \right) \frac{dy}{dx} = \frac{1}{x}\] ?
If \[x = \cos t \text{ and y } = \sin t,\] prove that \[\frac{dy}{dx} = \frac{1}{\sqrt{3}} \text { at } t = \frac{2 \pi}{3}\] ?
Differentiate log (1 + x2) with respect to tan−1 x ?
Differentiate \[\sin^{- 1} \left( 4x \sqrt{1 - 4 x^2} \right)\] with respect to \[\sqrt{1 - 4 x^2}\] , if \[x \in \left( \frac{1}{2 \sqrt{2}}, \frac{1}{2} \right)\] ?
If \[f\left( 0 \right) = f\left( 1 \right) = 0, f'\left( 1 \right) = 2 \text { and y } = f \left( e^x \right) e^{f \left( x \right)}\] write the value of \[\frac{dy}{dx} \text{ at x } = 0\] ?
If \[y = \sin^{- 1} \left( \frac{1 - x^2}{1 + x^2} \right) + \cos^{- 1} \left( \frac{1 - x^2}{1 + x^2} \right),\text{ find } \frac{dy}{dx}\] ?
If \[u = \sin^{- 1} \left( \frac{2x}{1 + x^2} \right) \text{ and v} = \tan^{- 1} \left( \frac{2x}{1 - x^2} \right)\] where \[- 1 < x < 1\], then write the value of \[\frac{du}{dv}\] ?
If \[y = \log \left| 3x \right|, x \neq 0, \text{ find } \frac{dy}{dx} \] ?
If f (x) is an odd function, then write whether `f' (x)` is even or odd ?
If \[\sin y = x \sin \left( a + y \right), \text { then }\frac{dy}{dx} \text { is}\] ____________ .
The derivative of \[\cos^{- 1} \left( 2 x^2 - 1 \right)\] with respect to \[\cos^{- 1} x\] is ___________ .
If \[f\left( x \right) = \left| x^2 - 9x + 20 \right|\] then `f' (x)` is equal to ____________ .
Find the second order derivatives of the following function x3 + tan x ?
If x = 4z2 + 5, y = 6z2 + 7z + 3, find \[\frac{d^2 y}{d x^2}\] ?
If y = a xn + 1 + bx−n and \[x^2 \frac{d^2 y}{d x^2} = \lambda y\] then write the value of λ ?
If \[y^\frac{1}{n} + y^{- \frac{1}{n}} = 2x, \text { then find } \left( x^2 - 1 \right) y_2 + x y_1 =\] ?
Differentiate the following with respect to x:
\[\cot^{- 1} \left( \frac{1 - x}{1 + x} \right)\]
Range of 'a' for which x3 – 12x + [a] = 0 has exactly one real root is (–∞, p) ∪ [q, ∞), then ||p| – |q|| is ______.
