Advertisements
Advertisements
Question
Differentiate \[\cos^{- 1} \left\{ \sqrt{\frac{1 + x}{2}} \right\}, - 1 < x < 1\] ?
Advertisements
Solution
\[\text{ Let, y }= \cos^{- 1} \left\{ \sqrt{\frac{1 + x}{2}} \right\}\]
\[\text{Put x } = \cos2\theta\]
\[ y = \cos^{- 1} \left\{ \sqrt{\frac{1 + \cos 2\theta}{2}} \right\}\]
\[ y = \cos^{- 1} \left\{ \sqrt{\frac{2 \cos^2 \theta}{2}} \right\}\]
\[ y = \cos^{- 1} \left( \cos\theta \right) . . . \left( i \right)\]
\[\text{ Here }, - 1 < x < 1\]
\[ \Rightarrow - 1 < \cos2\theta < 1\]
\[ \Rightarrow 0 < 2\theta < \pi\]
\[ \Rightarrow 0 < \theta < \frac{\pi}{2}\]
\[\text{ So, from equation } \left( i \right)\]
\[ y = \theta \left[ \text{ since } , \cos^{- 1} \left( cos\theta \right) = \theta, if\theta \in \left[ 0, \pi \right] \right]\]
\[ \Rightarrow y = \frac{1}{2} \cos^{- 1} x \left[ \text{ Since } , x = \cos2\theta \right]\]
\[\text{ Differentiating it with respect to x }, \]
\[\frac{d y}{d x} = - \frac{1}{2\sqrt{1 - x^2}}\]
APPEARS IN
RELATED QUESTIONS
Differentiate the following functions from first principles eax+b.
Differentiate the following functions from first principles log cos x ?
Differentiate tan (x° + 45°) ?
Differentiate \[\sqrt{\frac{1 - x^2}{1 + x^2}}\] ?
Differentiate \[e^{\tan 3 x} \] ?
Differentiate \[\tan^{- 1} \left( e^x \right)\] ?
Differentiate \[e^{\sin^{- 1} 2x}\] ?
Differentiate \[\sin^{- 1} \left( \frac{x}{\sqrt{x^2 + a^2}} \right)\] ?
\[\log\left\{ \cot\left( \frac{\pi}{4} + \frac{x}{2} \right) \right\}\] ?
Differentiate \[\log \sqrt{\frac{x - 1}{x + 1}}\] ?
If \[y = \frac{x \sin^{- 1} x}{\sqrt{1 - x^2}}\] , prove that \[\left( 1 - x^2 \right) \frac{dy}{dx} = x + \frac{y}{x}\] ?
Differentiate \[\cos^{- 1} \left\{ \frac{\cos x + \sin x}{\sqrt{2}} \right\}, - \frac{\pi}{4} < x < \frac{\pi}{4}\] ?
If \[y = \sin^{- 1} \left( \frac{x}{1 + x^2} \right) + \cos^{- 1} \left( \frac{1}{\sqrt{1 + x^2}} \right), 0 < x < \infty\] prove that \[\frac{dy}{dx} = \frac{2}{1 + x^2} \] ?
Find \[\frac{dy}{dx}\] in the following case \[4x + 3y = \log \left( 4x - 3y \right)\] ?
If \[\log \sqrt{x^2 + y^2} = \tan^{- 1} \left( \frac{y}{x} \right)\] Prove that \[\frac{dy}{dx} = \frac{x + y}{x - y}\] ?
Differentiate \[\left( \log x \right)^{ \log x }\] ?
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 \[y = x \sin \left( a + y \right)\] , prove that \[\frac{dy}{dx} = \frac{\sin^2 \left( a + y \right)}{\sin \left( a + y \right) - y \cos \left( a + y \right)}\] ?
If \[\left( \cos x \right)^y = \left( \cos y \right)^x , \text{ find } \frac{dy}{dx}\] ?
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 \[\sin^{- 1} \left( 4x \sqrt{1 - 4 x^2} \right)\] with respect to \[\sqrt{1 - 4 x^2}\] , if \[x \in \left( \frac{1}{2 \sqrt{2}}, \frac{1}{2} \right)\] ?
Differentiate \[\cos^{- 1} \left( 4 x^3 - 3x \right)\] with respect to \[\tan^{- 1} \left( \frac{\sqrt{1 - x^2}}{x} \right), \text{ if }\frac{1}{2} < x < 1\] ?
If \[\pi \leq x \leq 2\pi \text { and y } = \cos^{- 1} \left( \cos x \right), \text { find } \frac{dy}{dx}\] ?
If \[y = \sin^{- 1} \left( \frac{2x}{1 + x^2} \right)\] write the value of \[\frac{dy}{dx}\text { for } x > 1\] ?
If \[y = \log_a x, \text{ find } \frac{dy}{dx} \] ?
Find the second order derivatives of the following function tan−1 x ?
If y = e−x cos x, show that \[\frac{d^2 y}{d x^2} = 2 e^{- x} \sin x\] ?
If \[y = \left[ \log \left( x + \sqrt{x^2 + 1} \right) \right]^2\] show that \[\left( 1 + x^2 \right)\frac{d^2 y}{d x^2} + x\frac{dy}{dx} = 2\] ?
If y = cot x show that \[\frac{d^2 y}{d x^2} + 2y\frac{dy}{dx} = 0\] ?
If \[y = e^{a \cos^{- 1}} x\] ,prove that \[\left( 1 - x^2 \right)\frac{d^2 y}{d x^2} - x\frac{dy}{dx} - a^2 y = 0\] ?
If y = cosec−1 x, x >1, then show that \[x\left( x^2 - 1 \right)\frac{d^2 y}{d x^2} + \left( 2 x^2 - 1 \right)\frac{dy}{dx} = 0\] ?
\[\text { Find A and B so that y = A } \sin3x + B \cos3x \text { satisfies the equation }\]
\[\frac{d^2 y}{d x^2} + 4\frac{d y}{d x} + 3y = 10 \cos3x \] ?
If x = a cos nt − b sin nt and \[\frac{d^2 x}{dt} = \lambda x\] then find the value of λ ?
Let f(x) be a polynomial. Then, the second order derivative of f(ex) is
If y = etan x, then (cos2 x)y2 =
Find the minimum value of (ax + by), where xy = c2.
