Advertisements
Advertisements
प्रश्न
Differentiate \[x^{\sin^{- 1} x}\] ?
Advertisements
उत्तर
\[\text{ Let y } = x^{\sin^{- 1} x} . . . \left( i \right)\]
\[\text{ Taking log on both sides }, \]
\[\log y = \log x^{\sin^{- 1} x } \]
\[ \Rightarrow \log y = \sin^{- 1} x \log x \]
\[\text{ Differentiating with respect to x }, \]
\[\frac{1}{y}\frac{dy}{dx} = \sin^{- 1} x\frac{d}{dx}\left( \log x \right) + \left( \log x \right)\frac{d}{dx}\left( \sin^{- 1} x \right) \]
\[ \Rightarrow \frac{1}{y}\frac{dy}{dx} = \sin^{- 1} x\left( \frac{1}{x} \right) + \left( \log x \right)\left( \frac{1}{\sqrt{1 - x^2}} \right)\]
\[ \Rightarrow \frac{dy}{dx} = y\left[ \frac{\sin^{- 1} x}{x} + \frac{\log x}{\sqrt{1 - x^2}} \right]\]
\[ \Rightarrow \frac{dy}{dx} = x^{\sin^{- 1} x} \left[ \frac{\sin^{- 1} x}{x} + \frac{\log x}{\sqrt{1 - x^2}} \right] \left[ \text{ using equation } \left( i \right) \right]\]
APPEARS IN
संबंधित प्रश्न
Differentiate \[\frac{e^{2x} + e^{- 2x}}{e^{2x} - e^{- 2x}}\] ?
Differentiate \[\log \left( 3x + 2 \right) - x^2 \log \left( 2x - 1 \right)\] ?
Differentiate \[\frac{e^x \sin x}{\left( x^2 + 2 \right)^3}\] ?
If \[y = \sqrt{x + 1} + \sqrt{x - 1}\] , prove that \[\sqrt{x^2 - 1}\frac{dy}{dx} = \frac{1}{2}y\] ?
Differentiate \[\sin^{- 1} \left\{ \frac{x}{\sqrt{x^2 + a^2}} \right\}\] ?
Differentiate \[\sin^{- 1} \left( 1 - 2 x^2 \right), 0 < x < 1\] ?
Differentiate \[\tan^{- 1} \left( \frac{4x}{1 - 4 x^2} \right), - \frac{1}{2} < x < \frac{1}{2}\] ?
Differentiate \[\cos^{- 1} \left( \frac{1 - x^{2n}}{1 + x^{2n}} \right), < x < \infty\] ?
Differentiate
\[\tan^{- 1} \left( \frac{\cos x + \sin x}{\cos x - \sin x} \right), \frac{\pi}{4} < x < \frac{\pi}{4}\] ?
If \[y = \tan^{- 1} \left( \frac{\sqrt{1 + x} - \sqrt{1 - x}}{\sqrt{1 + x} + \sqrt{1 - x}} \right), \text{find } \frac{dy}{dx}\] ?
If \[\sin \left( xy \right) + \frac{y}{x} = x^2 - y^2 , \text{ find} \frac{dy}{dx}\] ?
If \[\sin^2 y + \cos xy = k,\] find \[\frac{dy}{dx}\] at \[x = 1 , \] \[y = \frac{\pi}{4} .\]
Differentiate \[{10}^\left( {10}^x \right)\] ?
Differentiate \[\left( \sin^{- 1} x \right)^x\] ?
Differentiate\[\left( x + \frac{1}{x} \right)^x + x^\left( 1 + \frac{1}{x} \right)\] ?
Find \[\frac{dy}{dx}\] \[y = \frac{e^{ax} \cdot \sec x \cdot \log x}{\sqrt{1 - 2x}}\] ?
If \[x^{13} y^7 = \left( x + y \right)^{20}\] prove that \[\frac{dy}{dx} = \frac{y}{x}\] ?
If \[\left( \sin x \right)^y = \left( \cos y \right)^x ,\], prove that \[\frac{dy}{dx} = \frac{\log \cos y - y cot x}{\log \sin x + x \tan y}\] ?
If \[\left( \sin x \right)^y = x + y\] , prove that \[\frac{dy}{dx} = \frac{1 - \left( x + y \right) y \cot x}{\left( x + y \right) \log \sin x - 1}\] ?
Find \[\frac{dy}{dx}\], When \[x = a \left( \theta + \sin \theta \right) \text{ and } y = a \left( 1 - \cos \theta \right)\] ?
Find \[\frac{dy}{dx}\] ,When \[x = a \left( 1 - \cos \theta \right) \text{ and } y = a \left( \theta + \sin \theta \right) \text{ at } \theta = \frac{\pi}{2}\] ?
Differentiate \[\sin^{- 1} \left( 2x \sqrt{1 - x^2} \right)\] with respect to \[\sec^{- 1} \left( \frac{1}{\sqrt{1 - x^2}} \right)\], if \[x \in \left( \frac{1}{\sqrt{2}}, 1 \right)\] ?
Differentiate \[\sin^{- 1} \left( \frac{2x}{1 + x^2} \right)\] with respect to \[\cos^{- 1} \left( \frac{1 - x^2}{1 + x^2} \right), \text { if } 0 < x < 1\] ?
Differentiate \[\sin^{- 1} \left( 2x \sqrt{1 - x^2} \right)\] with respect to \[\tan^{- 1} \left( \frac{x}{\sqrt{1 - x^2}} \right), \text { if }- \frac{1}{\sqrt{2}} < x < \frac{1}{\sqrt{2}}\] ?
Differentiate \[\tan^{- 1} \left( \frac{1 - x}{1 + x} \right)\] with respect to \[\sqrt{1 - x^2},\text {if} - 1 < x < 1\] ?
If \[f'\left( 1 \right) = 2 \text { and y } = f \left( \log_e x \right), \text { find} \frac{dy}{dx} \text { at }x = e\] ?
If \[\left| x \right| < 1 \text{ and y} = 1 + x + x^2 + . . \] to ∞, then find the value of \[\frac{dy}{dx}\] ?
Differential coefficient of sec(tan−1 x) is ______.
If \[f\left( x \right) = \tan^{- 1} \sqrt{\frac{1 + \sin x}{1 - \sin x}}, 0 \leq x \leq \pi/2, \text{ then } f' \left( \pi/6 \right) \text{ is }\] _________ .
If \[f\left( x \right) = \left| x^2 - 9x + 20 \right|\] then `f' (x)` is equal to ____________ .
If y = x + tan x, show that \[\cos^2 x\frac{d^2 y}{d x^2} - 2y + 2x = 0\] ?
If y = 2 sin x + 3 cos x, show that \[\frac{d^2 y}{d x^2} + y = 0\] ?
If x = t2 and y = t3, find \[\frac{d^2 y}{d x^2}\] ?
If x = f(t) and y = g(t), then write the value of \[\frac{d^2 y}{d x^2}\] ?
If y = a sin mx + b cos mx, then \[\frac{d^2 y}{d x^2}\] is equal to
\[\text { If } y = \left( x + \sqrt{1 + x^2} \right)^n , \text { then show that }\]
\[\left( 1 + x^2 \right)\frac{d^2 y}{d x^2} + x\frac{dy}{dx} = n^2 y .\]
Find the minimum value of (ax + by), where xy = c2.
If y = xx, prove that \[\frac{d^2 y}{d x^2} - \frac{1}{y} \left( \frac{dy}{dx} \right)^2 - \frac{y}{x} = 0 .\]
If x = a (1 + cos θ), y = a(θ + sin θ), prove that \[\frac{d^2 y}{d x^2} = \frac{- 1}{a}at \theta = \frac{\pi}{2}\]
