Advertisements
Advertisements
प्रश्न
The derivative of \[\cos^{- 1} \left( 2 x^2 - 1 \right)\] with respect to \[\cos^{- 1} x\] is ___________ .
विकल्प
`2`
\[\frac{1}{2 \sqrt{1 - x^2}}\]
\[2/x\]
\[1 - x^2\]
Advertisements
उत्तर
`2`
\[\text { Let u } = \cos^{- 1} \left( 2 x^2 - 1 \right)\]
\[\text { Put x } = \cos\theta\]
\[ \Rightarrow \theta = \cos^{- 1} x\]
\[\frac{d\theta}{dx} = \frac{- 1}{\sqrt{1 - x^2}}\]
\[\text { Now, u } = \cos^{- 1} \left( \cos2\theta \right)\]
\[ \Rightarrow u = 2\theta\]
\[\Rightarrow \frac{du}{dx} = 2\frac{d\theta}{dx}\]
\[ \Rightarrow \frac{du}{dx} = \frac{- 2}{\sqrt{1 - x^2}} . . . \left( i \right)\]
\[\text { and,} \]
\[ v = \cos^{- 1} x\]
\[ \Rightarrow v = \cos^{- 1} \left( \cos\theta \right)\]
\[ \Rightarrow v = \theta\]
\[\frac{dv}{dx} = \frac{d\theta}{dx}\]
\[ \Rightarrow \frac{dv}{dx} = \frac{- 1}{\sqrt{1 - x^2}} . . . \left( ii \right)\]
\[\text { Dividing } \left( i \right) \text { by }\left( ii \right), \text { we get }, \]
\[\frac{\frac{du}{dx}}{\frac{dv}{dx}} = \frac{- 2}{\sqrt{1 - x^2}} \times \frac{\sqrt{1 - x^2}}{- 1}\]
\[ \Rightarrow \frac{du}{dv} = 2\]
APPEARS IN
संबंधित प्रश्न
Differentiate \[e^{\sin^{- 1} 2x}\] ?
Differentiate \[\sqrt{\tan^{- 1} \left( \frac{x}{2} \right)}\] ?
Differentiate \[\frac{2^x \cos x}{\left( x^2 + 3 \right)^2}\]?
Differentiate \[\frac{3 x^2 \sin x}{\sqrt{7 - x^2}}\] ?
Differentiate \[\frac{x^2 + 2}{\sqrt{\cos x}}\] ?
Differentiate \[\cos \left( \log x \right)^2\] ?
If \[y = \frac{1}{2} \log \left( \frac{1 - \cos 2x }{1 + \cos 2x} \right)\] , prove that \[\frac{ dy }{ dx } = 2 \text{cosec }2x \] ?
If \[y = x \sin^{- 1} x + \sqrt{1 - x^2}\] ,prove that \[\frac{dy}{dx} = \sin^{- 1} x\] ?
Differentiate \[\sin^{- 1} \left\{ \frac{\sin x + \cos x}{\sqrt{2}} \right\}, - \frac{3 \pi}{4} < x < \frac{\pi}{4}\] ?
If \[y = se c^{- 1} \left( \frac{x + 1}{x - 1} \right) + \sin^{- 1} \left( \frac{x - 1}{x + 1} \right), x > 0 . \text{ Find} \frac{dy}{dx}\] ?
Differentiate \[x^{1/x}\] with respect to x.
Differentiate \[\left( \log x \right)^{\cos x}\] ?
Differentiate \[\left( \sin x \right)^{\cos x}\] ?
If `y=(sinx)^x + sin^-1 sqrtx "then find" dy/dx`
Find \[\frac{dy}{dx}\]
\[y = x^x + x^{1/x}\] ?
If \[xy = e^{x - y} , \text{ find } \frac{dy}{dx}\] ?
If \[y^x + x^y + x^x = a^b\] ,find \[\frac{dy}{dx}\] ?
Find \[\frac{dy}{dx}\], when \[x = a \left( \cos \theta + \theta \sin \theta \right) \text{ and }y = a \left( \sin \theta - \theta \cos \theta \right)\] ?
If \[x = a \left( \frac{1 + t^2}{1 - t^2} \right) \text { and y } = \frac{2t}{1 - t^2}, \text { find } \frac{dy}{dx}\] ?
If \[x = a \left( \theta - \sin \theta \right) and, y = a \left( 1 + \cos \theta \right), \text { find } \frac{dy}{dx} \text{ at }\theta = \frac{\pi}{3} \] ?
If \[x = \frac{1 + \log t}{t^2}, y = \frac{3 + 2\log t}{t}, \text { find } \frac{dy}{dx}\] ?
\[\sin^{- 1} \sqrt{1 - x^2}\] with respect to \[\cot^{- 1} \left( \frac{x}{\sqrt{1 - x^2}} \right),\text { if }0 < x < 1\] ?
Let g (x) be the inverse of an invertible function f (x) which is derivable at x = 3. If f (3) = 9 and `f' (3) = 9`, write the value of `g' (9)`.
If \[y = \sin^{- 1} \left( \frac{2x}{1 + x^2} \right)\] write the value of \[\frac{dy}{dx}\text { for } x > 1\] ?
If \[y = \sin^{- 1} x + \cos^{- 1} x\] ,find \[\frac{dy}{dx}\] ?
If \[y = \sqrt{\sin x + y}, \text { then }\frac{dy}{dx} \text { equals }\] ______________ .
If y = log (sin x), prove that \[\frac{d^3 y}{d x^3} = 2 \cos \ x \ {cosec}^3 x\] ?
If \[y = e^{2x} \left( ax + b \right)\] show that \[y_2 - 4 y_1 + 4y = 0\] ?
If y = (cot−1 x)2, prove that y2(x2 + 1)2 + 2x (x2 + 1) y1 = 2 ?
If y = cosec−1 x, x >1, then show that \[x\left( x^2 - 1 \right)\frac{d^2 y}{d x^2} + \left( 2 x^2 - 1 \right)\frac{dy}{dx} = 0\] ?
\[\text { If }y = A e^{- kt} \cos\left( pt + c \right), \text { prove that } \frac{d^2 y}{d t^2} + 2k\frac{d y}{d t} + n^2 y = 0, \text { where } n^2 = p^2 + k^2 \] ?
If x = at2, y = 2 at, then \[\frac{d^2 y}{d x^2} =\]
If xy − loge y = 1 satisfies the equation \[x\left( y y_2 + y_1^2 \right) - y_2 + \lambda y y_1 = 0\]
If y = xx, prove that \[\frac{d^2 y}{d x^2} - \frac{1}{y} \left( \frac{dy}{dx} \right)^2 - \frac{y}{x} = 0 .\]
f(x) = 3x2 + 6x + 8, x ∈ R
Find the height of a cylinder, which is open at the top, having a given surface area, greatest volume, and radius r.
