Advertisements
Advertisements
प्रश्न
Differentiate the following functions from first principles sin−1 (2x + 3) ?
Advertisements
उत्तर
\[\text{Let} f\left( x \right) = \sin^{- 1} \left( 2x + 3 \right)\]
\[ \Rightarrow f\left( x + h \right) = \sin^{- 1} \left( 2\left( x + h \right) + 3 \right)\]
\[ \Rightarrow f\left( x + h \right) = \sin^{- 1} \left( 2x + 2h + 3 \right)\]
\[ \therefore \frac{d}{dx}\left\{ f\left( x \right) \right\} = \lim_{h \to 0} \frac{f\left( x + h \right) - f\left( x \right)}{h}\]
\[ = \lim_{h \to 0} \frac{\sin^{- 1} \left( 2x + 2h + 3 \right) - \sin^{- 1} \left( 2x + 3 \right)}{h}\]
\[ = \lim_{h \to 0} \frac{\sin^{- 1} \left[ \left( 2x + 2h + 3 \right)\sqrt{1 - \left( 2x + 3 \right)^2} - \left( 2x + 3 \right)\sqrt{1 - \left( 2x + 2h + 3 \right)^2} \right]}{h} \left[ \because \sin^{- 1} x - \sin^{- 1} y = \sin^{- 1} \left[ x\sqrt{1 - y^2} - y\sqrt{1 - x^2} \right] \right]\]
\[ = \lim_{h \to 0} \frac{\sin^{- 1} z}{z} \times \frac{z}{h}\]
\[\text{where, } z = \left( 2x + 2h + 3 \right)\sqrt{1 - \left( 2x + 3 \right)^2} - \left( 2x + 3 \right)\sqrt{1 - \left( 2x + 2h + 3 \right)^2} \text{ and }\lim_{h \to 0} \frac{\sin^{- 1} h}{h} = 1\]
\[ = \lim_{h \to 0} \frac{z}{h}\]
\[ = \lim_{h \to 0} \frac{\left( 2x + 2h + 3 \right)\sqrt{1 - \left( 2x + 3 \right)^2} - \left( 2x + 3 \right)\sqrt{1 - \left( 2x + 2h + 3 \right)^2}}{h}\]
\[ = \lim_{h \to 0} \frac{\left( 2x + 2h + 3 \right)^2 \left\{ 1 - \left( 2x + 3 \right)^2 \right\} - \left( 2x + 3 \right)^2 \left\{ 1 - \left( 2x + 2h + 3 \right)^2 \right\}}{h\left\{ \left( 2x + 2h + 3 \right)\sqrt{1 - \left( 2x + 3 \right)^2} + \left( 2x + 3 \right)\sqrt{1 - \left( 2x + 2h + 3 \right)^2} \right\}} \] ...........[Rationalizing numerator]
\[ = \lim_{h \to 0} \frac{\left[ \left( 2x + 3 \right)^2 + 4 h^2 + 4h\left( 2x + 3 \right) \right]\left\{ 1 - \left( 2x + 3 \right)^2 \right\} - \left( 2x + 3 \right)^2 \left[ 1 - \left( 2x + 3 \right)^2 - 4 h^2 - 4h\left( 2x + 3 \right) \right]}{h\left\{ \left( 2x + 2h + 3 \right)\sqrt{1 - \left( 2x + 3 \right)^2} + \left( 2x + 3 \right)\sqrt{1 - \left( 2x + 2h + 3 \right)^2} \right\}}\]
\[ = \lim_{h \to 0} \frac{\left[ \left( 2x + 3 \right)^2 + 4 h^2 + 4h\left( 2x + 3 \right) - \left( 2x + 3 \right)^4 - 4 h^2 \left( 2x + 3 \right)^2 - 4h \left( 2x + 3 \right)^3 - \left( 2x + 3 \right)^2 + \left( 2x + 3 \right)^4 + 4 h^2 \left( 2x + 3 \right)^2 + 4h \left( 2x + 3 \right)^3 \right]}{h\left\{ \left( 2x + 2h + 3 \right)\sqrt{1 - \left( 2x + 3 \right)^2} + \left( 2x + 3 \right)\sqrt{1 - \left( 2x + 2h + 3 \right)^2} \right\}}\]
\[ = \lim_{h \to 0} \frac{4h\left[ h + \left( 2x + 3 \right) \right]}{h\left\{ \left( 2x + 2h + 3 \right)\sqrt{1 - \left( 2x + 3 \right)^2} + \left( 2x + 3 \right)\sqrt{1 - \left( 2x + 2h + 3 \right)^2} \right\}}\]
\[ = \frac{4\left( 2x + 3 \right)}{\left( 2x + 3 \right)\sqrt{1 - \left( 2x + 3 \right)^2} + \left( 2x + 3 \right)\sqrt{1 - \left( 2x + 3 \right)^2}}\]
\[ = \frac{4\left( 2x + 3 \right)}{2\left( 2x + 3 \right)\sqrt{1 - \left( 2x + 3 \right)^2}}\]
\[ = \frac{2}{\sqrt{1 - \left( 2x + 3 \right)^2}}\]
\[ \therefore \frac{d}{dx}\left\{ \sin^{- 1} \left( 2x + 3 \right) \right\} = \frac{2}{\sqrt{1 - \left( 2x + 3 \right)^2}}\]
APPEARS IN
संबंधित प्रश्न
Differentiate (log sin x)2 ?
Differentiate \[\sin \left( \frac{1 + x^2}{1 - x^2} \right)\] ?
Differentiate \[e^{\tan^{- 1}} \sqrt{x}\] ?
Differentiate \[\log \left( 3x + 2 \right) - x^2 \log \left( 2x - 1 \right)\] ?
Differentiate \[e^x \log \sin 2x\] ?
Differentiate \[\frac{x^2 + 2}{\sqrt{\cos x}}\] ?
If \[y = \sqrt{x + 1} + \sqrt{x - 1}\] , prove that \[\sqrt{x^2 - 1}\frac{dy}{dx} = \frac{1}{2}y\] ?
Differentiate \[\cos^{- 1} \left\{ 2x\sqrt{1 - x^2} \right\}, \frac{1}{\sqrt{2}} < x < 1\] ?
Differentiate \[\tan^{- 1} \left\{ \frac{x}{\sqrt{a^2 - x^2}} \right\}, - a < x < a\] ?
Differentiate \[\sin^{- 1} \left\{ \frac{x}{\sqrt{x^2 + a^2}} \right\}\] ?
Differentiate \[\tan^{- 1} \left( \frac{2^{x + 1}}{1 - 4^x} \right), - \infty < x < 0\] ?
Differentiate \[\tan^{- 1} \left( \frac{2 a^x}{1 - a^{2x}} \right), a > 1, - \infty < x < 0\] ?
Differentiate \[\tan^{- 1} \left( \frac{x}{1 + 6 x^2} \right)\] ?
Differentiate \[\tan^{- 1} \left\{ \frac{x^{1/3} + a^{1/3}}{1 - \left( a x \right)^{1/3}} \right\}\] ?
If \[y = \sin \left[ 2 \tan^{- 1} \left\{ \frac{\sqrt{1 - x}}{1 + x} \right\} \right], \text{ find } \frac{dy}{dx}\] ?
Find \[\frac{dy}{dx}\] in the following case \[x^5 + y^5 = 5 xy\] ?
If \[x y^2 = 1,\] prove that \[2\frac{dy}{dx} + y^3 = 0\] ?
If \[\tan \left( x + y \right) + \tan \left( x - y \right) = 1, \text{ find} \frac{dy}{dx}\] ?
If \[\sin^2 y + \cos xy = k,\] find \[\frac{dy}{dx}\] at \[x = 1 , \] \[y = \frac{\pi}{4} .\]
Differentiate \[\left( \sin^{- 1} x \right)^x\] ?
Find \[\frac{dy}{dx}\] \[y = \left( \tan x \right)^{\log x} + \cos^2 \left( \frac{\pi}{4} \right)\] ?
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 \[e^x + e^y = e^{x + y}\] , prove that
\[\frac{dy}{dx} + e^{y - x} = 0\] ?
If \[\left( \sin x \right)^y = x + y\] , prove that \[\frac{dy}{dx} = \frac{1 - \left( x + y \right) y \cot x}{\left( x + y \right) \log \sin x - 1}\] ?
If \[\frac{dy}{dx}\] when \[x = a \cos \theta \text{ and } y = b \sin \theta\] ?
If \[x = \frac{\sin^3 t}{\sqrt{\cos 2 t}}, y = \frac{\cos^3 t}{\sqrt{\cos t 2 t}}\] , find\[\frac{dy}{dx}\] ?
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 \[\tan^{- 1} \left( \frac{x}{\sqrt{1 - x^2}} \right)\] with respect to \[\sin^{- 1} \left( 2x \sqrt{1 - x^2} \right), \text { if } - \frac{1}{\sqrt{2}} < x < \frac{1}{\sqrt{2}}\] ?
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 \[\sin \left( x + y \right) = \log \left( x + y \right), \text { then } \frac{dy}{dx} =\] ___________ .
If \[3 \sin \left( xy \right) + 4 \cos \left( xy \right) = 5, \text { then } \frac{dy}{dx} =\] _____________ .
Find the second order derivatives of the following function x3 + tan x ?
If x = a (θ − sin θ), y = a (1 + cos θ) prove that, find \[\frac{d^2 y}{d x^2}\] ?
If \[y = \left[ \log \left( x + \sqrt{x^2 + 1} \right) \right]^2\] show that \[\left( 1 + x^2 \right)\frac{d^2 y}{d x^2} + x\frac{dy}{dx} = 2\] ?
\[ \text { If x } = a \sin t \text { and y } = a\left( \cos t + \log \tan\frac{t}{2} \right), \text { find } \frac{d^2 y}{d x^2} \] ?
If x = t2 and y = t3, find \[\frac{d^2 y}{d x^2}\] ?
If y = a sin mx + b cos mx, then \[\frac{d^2 y}{d x^2}\] is equal to
If x = f(t) cos t − f' (t) sin t and y = f(t) sin t + f'(t) cos t, then\[\left( \frac{dx}{dt} \right)^2 + \left( \frac{dy}{dt} \right)^2 =\]
