Advertisements
Advertisements
Question
Differentiate \[\sin^{- 1} \left\{ \frac{x}{\sqrt{x^2 + a^2}} \right\}\] ?
Advertisements
Solution
\[\text{ Let, y }= \sin^{- 1} \left\{ \frac{x}{\sqrt{x^2 + a^2}} \right\}\]
\[\text{put x } = a \tan\theta\]
\[ \Rightarrow y = \sin^{- 1} \left\{ \frac{a \tan\theta}{\sqrt{a^2 \tan^2 \theta + a^2}} \right\}\]
\[ \Rightarrow y = \sin^{- 1} \left\{ \frac{a \tan\theta}{\sqrt{a^2 \left( \tan^2 \theta + 1 \right)}} \right\}\]
\[ \Rightarrow y = \sin^{- 1} \left( \frac{a \tan\theta}{a sec\theta} \right) \]
\[ \Rightarrow y = \sin^{- 1} \left( \sin\theta \right) \]
\[ \Rightarrow y = \theta\]
\[ \Rightarrow y = \tan^{- 1} \left( \frac{x}{a} \right) \left[ \text{since, }x = a \tan\theta \right] \]
\[\text{ Differentiating it with respect to x using chain rule }, \]
\[ \frac{d y}{d x} = \frac{1}{1 + \left( \frac{x}{a} \right)^2}\frac{d}{dx}\left( \frac{x}{a} \right)\]
\[ \Rightarrow \frac{d y}{d x} = \frac{a^2}{a^2 + x^2} \times \left( \frac{1}{a} \right)\]
\[ \therefore \frac{d y}{d x} = \frac{a}{a^2 + x^2}\]
APPEARS IN
RELATED QUESTIONS
Differentiate tan2 x ?
Differentiate tan (x° + 45°) ?
Differentiate \[3^{e^x}\] ?
Differentiate \[\log \left( \frac{\sin x}{1 + \cos x} \right)\] ?
Differentiate \[\tan \left( e^{\sin x }\right)\] ?
Differentiate \[\frac{e^{2x} + e^{- 2x}}{e^{2x} - e^{- 2x}}\] ?
Differentiate \[\cos^{- 1} \left( \frac{x + \sqrt{1 - x^2}}{\sqrt{2}} \right), - 1 < x < 1\] ?
Differentiate \[\tan^{- 1} \left( \frac{2^{x + 1}}{1 - 4^x} \right), - \infty < x < 0\] ?
Differentiate \[\tan^{- 1} \left( \frac{a + x}{1 - ax} \right)\] ?
Find \[\frac{dy}{dx}\] in the following case \[x^5 + y^5 = 5 xy\] ?
If \[\tan^{- 1} \left( \frac{x^2 - y^2}{x^2 + y^2} \right) = a\] Prove that \[\frac{dy}{dx} = \frac{x}{y}\frac{\left( 1 - \tan a \right)}{\left( 1 + \tan a \right)}\] ?
If \[\tan \left( x + y \right) + \tan \left( x - y \right) = 1, \text{ find} \frac{dy}{dx}\] ?
Differentiate \[x^{1/x}\] with respect to x.
Differentiate \[\left( \sin x \right)^{\cos x}\] ?
Differentiate \[\left( \tan x \right)^{1/x}\] ?
Differentiate \[\left( x^x \right) \sqrt{x}\] ?
Differentiate \[x^\left( \sin x - \cos x \right) + \frac{x^2 - 1}{x^2 + 1}\] ?
Find \[\frac{dy}{dx}\] \[y = \sin x \sin 2x \sin 3x \sin 4x\] ?
Find \[\frac{dy}{dx}\] \[y = x^{\cos x} + \left( \sin x \right)^{\tan x}\] ?
If \[y = \sin \left( x^x \right)\] prove that \[\frac{dy}{dx} = \cos \left( x^x \right) \cdot x^x \left( 1 + \log x \right)\] ?
If \[y = \sqrt{x + \sqrt{x + \sqrt{x + . . . to \infty ,}}}\] prove that \[\frac{dy}{dx} = \frac{1}{2 y - 1}\] ?
If \[y = \left( \cos x \right)^{\left( \cos x \right)^{\left( \cos x \right) . . . \infty}}\],prove that \[\frac{dy}{dx} = - \frac{y^2 \tan x}{\left( 1 - y \log \cos x \right)}\]?
Find \[\frac{dy}{dx}\], When \[x = a \left( \theta + \sin \theta \right) \text{ and } y = a \left( 1 - \cos \theta \right)\] ?
If \[x = e^{\cos 2 t} \text{ and y }= e^{\sin 2 t} ,\] prove that \[\frac{dy}{dx} = - \frac{y \log x}{x \log y}\] ?
If \[f'\left( x \right) = \sqrt{2 x^2 - 1} \text { and y } = f \left( x^2 \right)\] then find \[\frac{dy}{dx} \text { at } x = 1\] ?
If \[x = 3\sin t - \sin3t, y = 3\cos t - \cos3t \text{ find }\frac{dy}{dx} \text{ at } t = \frac{\pi}{3}\] ?
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 \[y = \tan^{- 1} \left( \frac{\sin x + \cos x}{\cos x - \sin x} \right), \text { then } \frac{dy}{dx}\] is equal to ___________ .
If y = 2 sin x + 3 cos x, show that \[\frac{d^2 y}{d x^2} + y = 0\] ?
If x = a cos θ, y = b sin θ, show that \[\frac{d^2 y}{d x^2} = - \frac{b^4}{a^2 y^3}\] ?
If \[y = \tan^{- 1} \left\{ \frac{\log_e \left( e/ x^2 \right)}{\log_e \left( e x^2 \right)} \right\} + \tan^{- 1} \left( \frac{3 + 2 \log_e x}{1 - 6 \log_e x} \right)\], then \[\frac{d^2 y}{d x^2} =\]
If \[y^\frac{1}{n} + y^{- \frac{1}{n}} = 2x, \text { then find } \left( x^2 - 1 \right) y_2 + x y_1 =\] ?
If \[\frac{d}{dx}\left[ x^n - a_1 x^{n - 1} + a_2 x^{n - 2} + . . . + \left( - 1 \right)^n a_n \right] e^x = x^n e^x\] then the value of ar, 0 < r ≤ n, 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 `x=a (cos t +t sint )and y= a(sint-cos t )` Prove that `Sec^3 t/(at),0<t< pi/2`
Differentiate `log [x+2+sqrt(x^2+4x+1)]`
