Advertisements
Advertisements
प्रश्न
Differentiate \[\sin^{- 1} \left( 4x \sqrt{1 - 4 x^2} \right)\] with respect to \[\sqrt{1 - 4 x^2}\] , if \[x \in \left( - \frac{1}{2}, - \frac{1}{2 \sqrt{2}} \right)\] ?
उत्तर
\[\text { Let, u } = \sin^{- 1} \left( 4x\sqrt{1 - 4 x^2} \right)\]
\[ \text { put,} 2x = \cos\theta\]
\[ \Rightarrow u = \sin^{- 1} \left( 2 \times \cos\theta\sqrt{1 - \cos^2 \theta} \right)\]
\[ \Rightarrow u = \sin^{- 1} \left( 2\cos\theta \sin\theta \right) \]
\[ \Rightarrow u = \sin^{- 1} \left( \sin 2\theta \right) . . . \left( i \right)\]
\[ \text {Let, v } = \sqrt{1 - 4 x^2} . . . \left( ii \right)\]
\[\text { Here }, \]
\[ x \in \left( - \frac{1}{2}, - \frac{1}{2\sqrt{2}} \right)\]
\[ \Rightarrow 2x \in \left( - 1, - \frac{1}{\sqrt{2}} \right)\]
\[ \Rightarrow \theta \in \left( \frac{3\pi}{4}, \pi \right)\]
\[\text { So, from equation } \left( i \right), \]
\[ u = \pi - 2\theta ......\left[ \text { Since }, \sin^{- 1} \left( \sin\theta \right) = \pi - \theta , \text{ if }\theta \in \left( \frac{\pi}{2}, \frac{3\pi}{2} \right) \right]\]
\[ \Rightarrow u = \pi - 2 \cos^{- 1} \left( 2x \right) ........\left[ \text { Since }, 2x = \cos\theta \right]\]
Differentiate it with respect to x,
\[\frac{du}{dx} = 0 - 2\left( \frac{- 1}{\sqrt{1 - \left( 2x \right)^2}} \right)\frac{d}{dx}\left( 2x \right)\]
\[ \Rightarrow \frac{du}{dx} = \frac{2}{\sqrt{1 - 4 x^2}}\left( 2 \right)\]
\[ \Rightarrow \frac{du}{dx} = \frac{4}{\sqrt{1 - 4 x^2}} . . . \left( iii \right)\]
\[\text { from equation } \left( ii \right), \]
\[\frac{dv}{dx} = \frac{- 4x}{\sqrt{1 - 4 x^2}}\]
\[\text { but }, x \in \left( - \frac{1}{2}, - \frac{1}{2\sqrt{2}} \right)\]
\[ \therefore \frac{dv}{dx} = \frac{- 4\left( - x \right)}{\sqrt{1 - 4 \left( - x \right)^2}}\]
\[ \Rightarrow \frac{dv}{dx} = \frac{4x}{\sqrt{1 - 4 x^2}} . . . \left( iv \right)\]
\[\text { Dividing equation } \left( iii \right) by \left( iv \right)\]
\[\frac{\frac{du}{dx}}{\frac{dv}{dx}} = \frac{4}{\sqrt{1 - 4 x^2}} \times \frac{\sqrt{1 - 4 x^2}}{4x}\]
\[ \therefore \frac{du}{dv} = \frac{1}{x}\]
APPEARS IN
संबंधित प्रश्न
Differentiate tan 5x° ?
Differentiate \[\log \left( x + \sqrt{x^2 + 1} \right)\] ?
Differentiate \[\frac{e^x \log x}{x^2}\] ?
Differentiate \[e^{ax} \sec x \tan 2x\] ?
If \[y = e^x + e^{- x}\] prove that \[\frac{dy}{dx} = \sqrt{y^2 - 4}\] ?
Prove that \[\frac{d}{dx} \left\{ \frac{x}{2}\sqrt{a^2 - x^2} + \frac{a^2}{2} \sin^{- 1} \frac{x}{a} \right\} = \sqrt{a^2 - x^2}\] ?
Differentiate \[\cos^{- 1} \left\{ \frac{x}{\sqrt{x^2 + a^2}} \right\}\] ?
Differentiate \[\tan^{- 1} \left\{ \frac{x}{a + \sqrt{a^2 - x^2}} \right\}, - a < x < a\] ?
Differentiate \[\sin^{- 1} \left( \frac{x + \sqrt{1 - x^2}}{\sqrt{2}} \right), - 1 < x < 1\] ?
Differentiate \[\tan^{- 1} \left( \frac{x - a}{x + a} \right)\] ?
If \[y = \cot^{- 1} \left\{ \frac{\sqrt{1 + \sin x} + \sqrt{1 - \sin x}}{\sqrt{1 + \sin x} - \sqrt{1 - \sin x}} \right\}\], show that \[\frac{dy}{dx}\] is independent of x. ?
If \[\sec \left( \frac{x + y}{x - y} \right) = a\] Prove that \[\frac{dy}{dx} = \frac{y}{x}\] ?
If \[\sqrt{y + x} + \sqrt{y - x} = c, \text {show that } \frac{dy}{dx} = \frac{y}{x} - \sqrt{\frac{y^2}{x^2} - 1}\] ?
Differentiate \[x^{\cos^{- 1} x}\] ?
Differentiate \[\left( \log x \right)^{\cos x}\] ?
Differentiate \[\left( \sin x \right)^{\log x}\] ?
Differentiate \[x^{x \cos x +} \frac{x^2 + 1}{x^2 - 1}\] ?
Find \[\frac{dy}{dx}\] \[y = \frac{e^{ax} \cdot \sec x \cdot \log x}{\sqrt{1 - 2x}}\] ?
If `y=(sinx)^x + sin^-1 sqrtx "then find" dy/dx`
If \[xy = e^{x - y} , \text{ find } \frac{dy}{dx}\] ?
Find \[\frac{dy}{dx}\] ,When \[x = e^\theta \left( \theta + \frac{1}{\theta} \right) \text{ and } y = e^{- \theta} \left( \theta - \frac{1}{\theta} \right)\] ?
If \[x = \cos t \text{ and y } = \sin t,\] prove that \[\frac{dy}{dx} = \frac{1}{\sqrt{3}} \text { at } t = \frac{2 \pi}{3}\] ?
If \[x = \left( t + \frac{1}{t} \right)^a , y = a^{t + \frac{1}{t}} , \text{ find } \frac{dy}{dx}\] ?
Differentiate \[\sin^{- 1} \left( 2x \sqrt{1 - x^2} \right)\] with respect to \[\sec^{- 1} \left( \frac{1}{\sqrt{1 - x^2}} \right)\], if \[x \in \left( \frac{1}{\sqrt{2}}, 1 \right)\] ?
Differentiate \[\left( \cos x \right)^{\sin x }\] with respect to \[\left( \sin x \right)^{\cos x }\]?
If \[\left| x \right| < 1 \text{ and y} = 1 + x + x^2 + . . \] to ∞, then find the value of \[\frac{dy}{dx}\] ?
If \[f\left( x \right) = \sqrt{x^2 - 10x + 25}\] then the derivative of f (x) in the interval [0, 7] is ____________ .
If \[\sqrt{1 - x^6} + \sqrt{1 - y^6} = a^3 \left( x^3 - y^3 \right)\] then \[\frac{dy}{dx}\] is equal to ____________ .
If \[\sin^{- 1} \left( \frac{x^2 - y^2}{x^2 + y^2} \right) = \text { log a then } \frac{dy}{dx}\] is equal to _____________ .
Find the second order derivatives of the following function sin (log x) ?
Find the second order derivatives of the following function ex sin 5x ?
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 x = at2, y = 2 at, then \[\frac{d^2 y}{d x^2} =\]
If x = 2 at, y = at2, where a is a constant, then \[\frac{d^2 y}{d x^2} \text { at x } = \frac{1}{2}\] is
If y = sin (m sin−1 x), then (1 − x2) y2 − xy1 is equal to
If \[y = \log_e \left( \frac{x}{a + bx} \right)^x\] then x3 y2 =
If y = xn−1 log x then x2 y2 + (3 − 2n) xy1 is equal to
\[\text { If } y = \left( x + \sqrt{1 + x^2} \right)^n , \text { then show that }\]
\[\left( 1 + x^2 \right)\frac{d^2 y}{d x^2} + x\frac{dy}{dx} = n^2 y .\]
