Advertisements
Advertisements
Question
Differentiate \[\cos^{- 1} \left( \frac{x + \sqrt{1 - x^2}}{\sqrt{2}} \right), - 1 < x < 1\] ?
Solution
\[\text{ Let y } = \cos^{- 1} \left\{ \frac{x + \sqrt{1 - x^2}}{\sqrt{2}} \right\}\]
\[\text{ Put, x } = \cos\theta\]
\[ y = \cos^{- 1} \left\{ \frac{\cos\theta + \sqrt{1 - \cos^2 \theta}}{\sqrt{2}} \right\}\]
\[ y = \cos^{- 1} \left\{ \frac{\cos\theta + \sin\theta}{\sqrt{2}} \right\}\]
\[ y = \cos^{- 1} \left\{ \cos\theta\left( \frac{1}{\sqrt{2}} \right) + \sin\theta\left( \frac{1}{\sqrt{2}} \right) \right\}\]
\[ y = \cos^{- 1} \left\{ \cos\theta\cos\frac{\pi}{4} + \sin\theta \sin\frac{\pi}{4} \right\}\]
\[ y = \cos^{- 1} \left\{ \cos\left( \theta - \frac{\pi}{4} \right) \right\} . . . \left( i \right)\]
\[\text{ Here }, - 1 < x < 1\]
\[ \Rightarrow - 1 < \cos\theta < 1 \]
\[ \Rightarrow \frac{3\pi}{4} < \theta < \frac{5\pi}{4} \]
\[ \Rightarrow \left( \frac{3\pi}{4} - \frac{\pi}{4} \right) < \left( \theta - \frac{\pi}{4} \right) < \frac{5\pi}{4} - \frac{\pi}{4}\]
\[ \Rightarrow \left( \frac{\pi}{2} \right) < \left( \theta - \frac{\pi}{4} \right) < \pi\]
\[\text{ So, from equation } \left( i \right), \]
\[ y = \left( \theta - \frac{\pi}{4} \right) ..........\left[ \text{ Since }, \cos^{- 1} \left( \cos\theta \right) = \theta, \text{ if }\theta \in \left[ 0, \pi \right] \right] \]
\[ y = \cos^{- 1} x - \frac{\pi}{4} ............\left[ \text{ Since }, x = \sin\theta \right]\]
\[\text{ Differentiating it with respect to x }, \]
\[\frac{d y}{d x} = - \frac{1}{\sqrt{1 - x^2}} + 0\]
\[\frac{d y}{d x} = - \frac{1}{\sqrt{1 - x^2}}\]
APPEARS IN
RELATED QUESTIONS
Differentiate \[\sqrt{\frac{1 + \sin x}{1 - \sin x}}\] ?
Differentiate \[e^{3 x} \cos 2x\] ?
Differentiate \[\tan^{- 1} \left( e^x \right)\] ?
Differentiate \[\sqrt{\tan^{- 1} \left( \frac{x}{2} \right)}\] ?
Differentiate \[\frac{2^x \cos x}{\left( x^2 + 3 \right)^2}\] ?
Differentiate \[x \sin 2x + 5^x + k^k + \left( \tan^2 x \right)^3\] ?
Differentiate \[e^x \log \sin 2x\] ?
Differentiate \[\left( \sin^{- 1} x^4 \right)^4\] ?
If \[y = \sqrt{x^2 + a^2}\] prove that \[y\frac{dy}{dx} - x = 0\] ?
Differentiate \[\tan^{- 1} \left( \frac{x - a}{x + a} \right)\] ?
If \[y = \tan^{- 1} \left( \frac{2x}{1 - x^2} \right) + \sec^{- 1} \left( \frac{1 + x^2}{1 - x^2} \right), x > 0\] ,prove that \[\frac{dy}{dx} = \frac{4}{1 + x^2} \] ?
If \[y = \cos^{- 1} \left\{ \frac{2x - 3 \sqrt{1 - x^2}}{\sqrt{13}} \right\}, \text{ find } \frac{dy}{dx}\] ?
Find \[\frac{dy}{dx}\] in the following case \[e^{x - y} = \log \left( \frac{x}{y} \right)\] ?
Differentiate \[\left( \log x \right)^{\cos x}\] ?
Find \[\frac{dy}{dx}\] \[y = x^{\cos x} + \left( \sin x \right)^{\tan x}\] ?
If \[x^{16} y^9 = \left( x^2 + y \right)^{17}\] ,prove that \[x\frac{dy}{dx} = 2 y\] ?
If \[y = x \sin \left( a + y \right)\] , prove that \[\frac{dy}{dx} = \frac{\sin^2 \left( a + y \right)}{\sin \left( a + y \right) - y \cos \left( a + y \right)}\] ?
If \[xy \log \left( x + y \right) = 1\] , prove that \[\frac{dy}{dx} = - \frac{y \left( x^2 y + x + y \right)}{x \left( x y^2 + x + y \right)}\] ?
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)` ?
Find \[\frac{dy}{dx}\] , when \[x = b \sin^2 \theta \text{ and } y = a \cos^2 \theta\] ?
Find \[\frac{dy}{dx}\] ,when \[x = \frac{e^t + e^{- t}}{2} \text{ and } y = \frac{e^t - e^{- t}}{2}\] ?
If \[f\left( 1 \right) = 4, f'\left( 1 \right) = 2\] find the value of the derivative of \[\log \left( f\left( e^x \right) \right)\] w.r. to x at the point x = 0 ?
If \[y = \sin^{- 1} \left( \frac{2x}{1 + x^2} \right)\] write the value of \[\frac{dy}{dx}\text { for } x > 1\] ?
If \[y = \sec^{- 1} \left( \frac{x + 1}{x - 1} \right) + \sin^{- 1} \left( \frac{x - 1}{x + 1} \right)\] then write the value of \[\frac{dy}{dx} \] ?
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 \[y = \left( 1 + \frac{1}{x} \right)^x , \text{ then} \frac{dy}{dx} =\] ____________ .
If \[x = a \cos^3 \theta, y = a \sin^3 \theta, \text { then } \sqrt{1 + \left( \frac{dy}{dx} \right)^2} =\] ____________ .
If \[f\left( x \right) = \sqrt{x^2 + 6x + 9}, \text { then } f'\left( x \right)\] is equal to ______________ .
If y = x3 log x, prove that \[\frac{d^4 y}{d x^4} = \frac{6}{x}\] ?
If y = log (sin x), prove that \[\frac{d^3 y}{d x^3} = 2 \cos \ x \ {cosec}^3 x\] ?
If x = a (θ − sin θ), y = a (1 + cos θ) prove that, find \[\frac{d^2 y}{d x^2}\] ?
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 ?
\[ \text { If x } = a \sin t \text { and y } = a\left( \cos t + \log \tan\frac{t}{2} \right), \text { find } \frac{d^2 y}{d x^2} \] ?
If \[y = 1 - x + \frac{x^2}{2!} - \frac{x^3}{3!} + \frac{x^4}{4!}\] .....to ∞, then write \[\frac{d^2 y}{d x^2}\] in terms of y ?
If \[y = \left| \log_e x \right|\] find\[\frac{d^2 y}{d x^2}\] ?
If y = sin (m sin−1 x), then (1 − x2) y2 − xy1 is equal to
Differentiate `log [x+2+sqrt(x^2+4x+1)]`
