Advertisements
Advertisements
प्रश्न
Differentiate \[\tan^{- 1} \left( \frac{2^{x + 1}}{1 - 4^x} \right), - \infty < x < 0\] ?
Advertisements
उत्तर
\[\text{ Let, y } = \tan^{- 1} \left\{ \frac{2^{x + 1}}{1 - 4^x} \right\}\]
\[\text{ put }2^x = \tan\theta\]
\[ y = \tan^{- 1} \left\{ \frac{2^x \times 2}{1 - \left( 2^x \right)^2} \right\}\]
\[ y = \tan^{- 1} \left( \frac{2 \tan\theta}{1 - \tan^2 \theta} \right) \]
\[ y = \tan^{- 1} \left( \tan2\theta \right) . . . \left( i \right)\]
\[\text{ Here }, - \infty < x < 0\]
\[ \Rightarrow 2^{- \infty} < 2^x < 2^\circ\]
\[ \Rightarrow 0 < 2^x < 1\]
\[ \Rightarrow 0 < \theta < \frac{\pi}{4}\]
\[ \Rightarrow 0 < 2\theta < \frac{\pi}{2}\]
\[\text{ So, from equation } \left( i \right), \]
\[ y = 2\theta ............\left[ \text{ Since }, \tan^{- 1} \left( \tan\theta \right) = \theta, \text{ if }\theta \in \left( - \frac{\pi}{2}, \frac{\pi}{2} \right) \right]\]
\[ y = 2 \tan^{- 1} \left( 2^x \right) \]
\[\text{ Differentiating it with respect to x} , \]
\[\frac{d y}{d x} = \frac{2}{1 + \left( 2^x \right)^2}\frac{d}{dx}\left( 2^x \right)\]
\[\frac{d y}{d x} = \frac{2 \times 2^x \log_e 2}{1 + 4^x}\]
\[\frac{d y}{d x} = \frac{2^{x + 1} \log_e 2}{1 + 4^x}\]
APPEARS IN
संबंधित प्रश्न
Differentiate the following functions from first principles log cos x ?
Differentiate etan x ?
Differentiate \[3^{x \log x}\] ?
Differentiate \[\sin \left( \frac{1 + x^2}{1 - x^2} \right)\] ?
Differentiate \[e^{\sin^{- 1} 2x}\] ?
Differentiate \[\frac{x^2 \left( 1 - x^2 \right)}{\cos 2x}\] ?
If \[y = \frac{x}{x + 2}\] , prove tha \[x\frac{dy}{dx} = \left( 1 - y \right) y\] ?
If \[y = \sqrt{x^2 + a^2}\] prove that \[y\frac{dy}{dx} - x = 0\] ?
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 \[\sin^{- 1} \left( 1 - 2 x^2 \right), 0 < x < 1\] ?
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 \[\frac{dy}{dx} = \frac{2}{1 + x^2} \] ?
If \[y = \cos^{- 1} \left( 2x \right) + 2 \cos^{- 1} \sqrt{1 - 4 x^2}, 0 < x < \frac{1}{2}, \text{ find } \frac{dy}{dx} .\] ?
If the derivative of tan−1 (a + bx) takes the value 1 at x = 0, prove that 1 + a2 = b ?
If \[y = \cos^{- 1} \left( 2x \right) + 2 \cos^{- 1} \sqrt{1 - 4 x^2}, - \frac{1}{2} < x < 0, \text{ find } \frac{dy}{dx} \] ?
Differentiate \[\left( \log x \right)^{\cos x}\] ?
Differentiate \[\left( \sin x \right)^{\log x}\] ?
Differentiate \[\left( x^x \right) \sqrt{x}\] ?
Find \[\frac{dy}{dx}\] \[y = \left( \sin x \right)^{\cos x} + \left( \cos x \right)^{\sin x}\] ?
If \[x^m y^n = 1\] , prove that \[\frac{dy}{dx} = - \frac{my}{nx}\] ?
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 \[x = a\left( t + \frac{1}{t} \right) \text{ and y } = a\left( t - \frac{1}{t} \right)\] ,prove that \[\frac{dy}{dx} = \frac{x}{y}\]?
Differentiate log (1 + x2) with respect to tan−1 x ?
Differentiate \[\tan^{- 1} \left( \frac{2x}{1 - x^2} \right)\] with respect to \[\cos^{- 1} \left( \frac{1 - x^2}{1 + x^2} \right),\text { if }0 < x < 1\] ?
If \[y = \sin^{- 1} x + \cos^{- 1} x\] ,find \[\frac{dy}{dx}\] ?
If \[- \frac{\pi}{2} < x < 0 \text{ and y } = \tan^{- 1} \sqrt{\frac{1 - \cos 2x}{1 + \cos 2x}}, \text{ find } \frac{dy}{dx}\] ?
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 \[y = \left( 1 + \frac{1}{x} \right)^x , \text{then} \frac{dy}{dx} =\] ____________.
If \[3 \sin \left( xy \right) + 4 \cos \left( xy \right) = 5, \text { then } \frac{dy}{dx} =\] _____________ .
If \[y = \sqrt{\sin x + y}, \text { then }\frac{dy}{dx} \text { equals }\] ______________ .
Find the second order derivatives of the following function x cos x ?
If x = t2 and y = t3, find \[\frac{d^2 y}{d x^2}\] ?
If x = 2at, y = at2, where a is a constant, then find \[\frac{d^2 y}{d x^2} \text { at }x = \frac{1}{2}\] ?
If y = a sin mx + b cos mx, then \[\frac{d^2 y}{d x^2}\] is equal to
If xy = e(x – y), then show that `dy/dx = (y(x-1))/(x(y+1)) .`
If `x=a (cos t +t sint )and y= a(sint-cos t )` Prove that `Sec^3 t/(at),0<t< pi/2`
f(x) = xx has a stationary point at ______.
