Advertisements
Advertisements
प्रश्न
Differentiate \[\sin^{- 1} \left\{ \sqrt{1 - x^2} \right\}, 0 < x < 1\] ?
Advertisements
उत्तर
\[\text{ Let, y } = \sin^{- 1} \left\{ \sqrt{1 - x^2} \right\}\]
\[\text{ Put, x } = \cos \theta\]
\[ y = \sin^{- 1} \left\{ \sqrt{1 - \cos^2 \theta} \right\}\]
\[ y = \sin^{- 1} \left( \sin\theta \right) . . . \left( 1 \right)\]
\[\text{ Here }, 0 < x < 1\]
\[ \Rightarrow 0 < \cos \theta < 1\]
\[ \Rightarrow 0 < \theta < \frac{\pi}{2}\]
\[\text{ So, from equation} \left( 1 \right), \]
\[ y = \theta \left[ \text{Since }, \sin^{- 1} \left( \sin\theta \right) = \theta, \text{ if } \theta \in \left[ - \frac{\pi}{2}, \frac{\pi}{2} \right] \right]\]
\[ y = \cos^{- 1} x \left[ \text{ Since }, x = \cos \theta \right]\]
\[\text{ Differentiating it with respect to x }, \]
\[\frac{d y}{d x} = - \frac{1}{\sqrt{1 - x^2}}\]
APPEARS IN
संबंधित प्रश्न
Differentiate the following functions from first principles log cos x ?
Differentiate the following function from first principles \[e^\sqrt{\cot x}\] .
Differentiate the following functions from first principles sin−1 (2x + 3) ?
Differentiate sin (3x + 5) ?
Differentiate tan2 x ?
Differentiate `2^(x^3)` ?
Differentiate \[\sqrt{\frac{1 + \sin x}{1 - \sin x}}\] ?
Differentiate \[\frac{e^x \log x}{x^2}\] ?
Prove that \[\frac{d}{dx} \left\{ \frac{x}{2}\sqrt{a^2 - x^2} + \frac{a^2}{2} \sin^{- 1} \frac{x}{a} \right\} = \sqrt{a^2 - x^2}\] ?
Differentiate \[\sin^{- 1} \left\{ \frac{\sin x + \cos x}{\sqrt{2}} \right\}, - \frac{3 \pi}{4} < x < \frac{\pi}{4}\] ?
Differentiate \[\tan^{- 1} \left( \frac{a + bx}{b - ax} \right)\] ?
Find \[\frac{dy}{dx}\] in the following case \[xy = c^2\] ?
Find \[\frac{dy}{dx}\] in the following case \[e^{x - y} = \log \left( \frac{x}{y} \right)\] ?
If \[\sec \left( \frac{x + y}{x - y} \right) = a\] Prove that \[\frac{dy}{dx} = \frac{y}{x}\] ?
If \[\sin \left( xy \right) + \frac{y}{x} = x^2 - y^2 , \text{ find} \frac{dy}{dx}\] ?
If \[e^x + e^y = e^{x + y} , \text{ prove that } \frac{dy}{dx} = - \frac{e^x \left( e^y - 1 \right)}{e^y \left( e^x - 1 \right)} or \frac{dy}{dx} + e^{y - x} = 0\] ?
Differentiate \[x^{1/x}\] with respect to x.
If \[x \sin \left( a + y \right) + \sin a \cos \left( a + y \right) = 0\] , prove that \[\frac{dy}{dx} = \frac{\sin^2 \left( a + y \right)}{\sin a}\] ?
If \[y = \left( \tan x \right)^{\left( \tan x \right)^{\left( \tan x \right)^{. . . \infty}}}\], prove that \[\frac{dy}{dx} = 2\ at\ x = \frac{\pi}{4}\] ?
If \[y = e^{x^{e^x}} + x^{e^{e^x}} + e^{x^{x^e}}\], prove that \[\frac{dy}{dx} = e^{x^{e^x}} \cdot x^{e^x} \left\{ \frac{e^x}{x} + e^x \cdot \log x \right\}+ x^{e^{e^x}} \cdot e^{e^x} \left\{ \frac{1}{x} + e^x \cdot \log x \right\} + e^{x^{x^e}} x^{x^e} \cdot x^{e - 1} \left\{ x + e \log x \right\}\]
Find \[\frac{dy}{dx}\],when \[x = a e^\theta \left( \sin \theta - \cos \theta \right), y = a e^\theta \left( \sin \theta + \cos \theta \right)\] ?
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 \[\sin^{- 1} \left( 4x \sqrt{1 - 4 x^2} \right)\] with respect to \[\sqrt{1 - 4 x^2}\] , if \[x \in \left( \frac{1}{2 \sqrt{2}}, \frac{1}{2} \right)\] ?
\[\sin^{- 1} \sqrt{1 - x^2}\] with respect to \[\cot^{- 1} \left( \frac{x}{\sqrt{1 - x^2}} \right),\text { if }0 < x < 1\] ?
If \[y = \sin^{- 1} \left( \frac{2x}{1 + x^2} \right)\] write the value of \[\frac{dy}{dx}\text { for } x > 1\] ?
If \[y = \log \left| 3x \right|, x \neq 0, \text{ find } \frac{dy}{dx} \] ?
\[\frac{d}{dx} \left\{ \tan^{- 1} \left( \frac{\cos x}{1 + \sin x} \right) \right\} \text { equals }\] ______________ .
If \[\sqrt{1 - x^6} + \sqrt{1 - y^6} = a^3 \left( x^3 - y^3 \right)\] then \[\frac{dy}{dx}\] is equal to ____________ .
If \[y = \log \sqrt{\tan x}\] then the value of \[\frac{dy}{dx}\text { at }x = \frac{\pi}{4}\] is given by __________ .
If y = 2 sin x + 3 cos x, show that \[\frac{d^2 y}{d x^2} + y = 0\] ?
If y = tan−1 x, show that \[\left( 1 + x^2 \right) \frac{d^2 y}{d x^2} + 2x\frac{dy}{dx} = 0\] ?
If y = (tan−1 x)2, then prove that (1 + x2)2 y2 + 2x(1 + x2)y1 = 2 ?
If y = axn+1 + bx−n, then \[x^2 \frac{d^2 y}{d x^2} =\]
If \[y = \frac{ax + b}{x^2 + c}\] then (2xy1 + y)y3 =
If x = sin t and y = sin pt, prove that \[\left( 1 - x^2 \right)\frac{d^2 y}{d x^2} - x\frac{dy}{dx} + p^2 y = 0\] .
Find the height of a cylinder, which is open at the top, having a given surface area, greatest volume, and radius r.
