Advertisements
Advertisements
प्रश्न
If \[y = \tan^{- 1} \left( \frac{\sin x + \cos x}{\cos x - \sin x} \right), \text { then } \frac{dy}{dx}\] is equal to ___________ .
पर्याय
`1/2`
0
1
none of these
Advertisements
उत्तर
`1`
\[\text { We have, y } = \tan^{- 1} \left( \frac{\sin x + \cos x}{\cos x - \sin x} \right)\]
\[ \Rightarrow \frac{dy}{dx} = \frac{1}{1 + \left( \frac{\sin x + \cos x}{\cos x - \sin x} \right)^2}\frac{d}{dx}\left( \frac{\sin x + \cos x}{\cos x - \sin x} \right)\]
\[ \Rightarrow \frac{dy}{dx} = \frac{\left( \cos x - \sin x \right)^2}{\left( \cos x - \sin x \right)^2 + \left( \sin x + \cos x \right)^2}\left[ \frac{\left( \cos x - \sin x \right)\frac{d}{dx}\left( \sin x + \cos x \right) - \left( \sin x + \cos x \right)\frac{d}{dx}\left( \cos x - \sin x \right)}{\left( \cos x - \sin x \right)^2} \right]\]
\[ \Rightarrow \frac{dy}{dx} = \frac{\left( \cos x - \sin x \right)^2}{\left( \cos x - \sin x \right)^2 + \left( \sin x + \cos x \right)^2}\left[ \frac{\left( \cos x - \sin x \right)\left( \cos x - \sin x \right) - \left( \sin x + \cos x \right)\left( - \sin x - \cos x \right)}{\left( \cos x - \sin x \right)^2} \right]\]
\[ \Rightarrow \frac{dy}{dx} = \frac{\left( \cos x - \sin x \right)^2}{\left( \cos x - \sin x \right)^2 + \left( \sin x + \cos x \right)^2}\left[ \frac{\left( \cos x - \sin x \right)\left( \cos x - \sin x \right) + \left( \sin x + \cos x \right)\left( \sin x + \cos x \right)}{\left( \cos x - \sin x \right)^2} \right]\]
\[ \Rightarrow \frac{dy}{dx} = \frac{\left( \cos x - \sin x \right)^2}{\left( \cos x - \sin x \right)^2 + \left( \sin x + \cos x \right)^2} \times \frac{\left( \cos x - \sin x \right)^2 + \left( \sin x + \cos x \right)^2}{\left( \cos x - \sin x \right)^2}\]
\[ \Rightarrow \frac{dy}{dx} = 1\]
APPEARS IN
संबंधित प्रश्न
Differentiate tan (x° + 45°) ?
Differentiate etan x ?
Differentiate tan 5x° ?
Differentiate \[e^\sqrt{\cot x}\] ?
Differentiate \[\log \sqrt{\frac{1 - \cos x}{1 + \cos x}}\] ?
Differentiate \[\frac{e^{2x} + e^{- 2x}}{e^{2x} - e^{- 2x}}\] ?
Differentiate \[\frac{2^x \cos x}{\left( x^2 + 3 \right)^2}\]?
If \[y = \log \left\{ \sqrt{x - 1} - \sqrt{x + 1} \right\}\] ,show that \[\frac{dy}{dx} = \frac{- 1}{2\sqrt{x^2 - 1}}\] ?
Differentiate \[\sin^{- 1} \left( 2 x^2 - 1 \right), 0 < x < 1\] ?
Differentiate \[\tan^{- 1} \left\{ \frac{x}{1 + \sqrt{1 - x^2}} \right\}, - 1 < x < 1\] ?
Differentiate \[\tan^{- 1} \left( \frac{\sin x}{1 + \cos x} \right), - \pi < x < \pi\] ?
Differentiate \[\tan^{- 1} \left( \frac{a + x}{1 - ax} \right)\] ?
Differentiate
\[\tan^{- 1} \left( \frac{\cos x + \sin x}{\cos x - \sin x} \right), \frac{\pi}{4} < x < \frac{\pi}{4}\] ?
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 \[\frac{dy}{dx}\] \[y = e^x + {10}^x + x^x\] ?
find \[\frac{dy}{dx}\] \[y = \frac{\left( x^2 - 1 \right)^3 \left( 2x - 1 \right)}{\sqrt{\left( x - 3 \right) \left( 4x - 1 \right)}}\] ?
Find \[\frac{dy}{dx}\] \[y = \frac{e^{ax} \cdot \sec x \cdot \log x}{\sqrt{1 - 2x}}\] ?
Find \[\frac{dy}{dx}\] \[y = e^{3x} \sin 4x \cdot 2^x\] ?
If \[x^{13} y^7 = \left( x + y \right)^{20}\] prove that \[\frac{dy}{dx} = \frac{y}{x}\] ?
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 \[x^x + y^x = 1\], prove that \[\frac{dy}{dx} = - \left\{ \frac{x^x \left( 1 + \log x \right) + y^x \cdot \log y}{x \cdot y^\left( x - 1 \right)} \right\}\] ?
If \[xy = e^{x - y} , \text{ find } \frac{dy}{dx}\] ?
Find \[\frac{dy}{dx}\],when \[x = a e^\theta \left( \sin \theta - \cos \theta \right), y = a e^\theta \left( \sin \theta + \cos \theta \right)\] ?
Differentiate log (1 + x2) with respect to tan−1 x ?
Differentiate (log x)x with respect to log x ?
If \[f\left( x \right) = \sqrt{x^2 - 10x + 25}\] then the derivative of f (x) in the interval [0, 7] is ____________ .
If x = a sec θ, y = b tan θ, prove that \[\frac{d^2 y}{d x^2} = - \frac{b^4}{a^2 y^3}\] ?
If y = 3 cos (log x) + 4 sin (log x), prove that x2y2 + xy1 + y = 0 ?
If log y = tan−1 x, show that (1 + x2)y2 + (2x − 1) y1 = 0 ?
If y = ex (sin x + cos x) prove that \[\frac{d^2 y}{d x^2} - 2\frac{dy}{dx} + 2y = 0\] ?
\[\text{ If x } = a\left( \cos t + \log \tan\frac{t}{2} \right) \text { and y } = a\left( \sin t \right), \text { evaluate } \frac{d^2 y}{d x^2} \text { at t } = \frac{\pi}{3} \] ?
\[\frac{d^{20}}{d x^{20}} \left( 2 \cos x \cos 3 x \right) =\]
If y = a + bx2, a, b arbitrary constants, then
If y = sin (m sin−1 x), then (1 − x2) y2 − xy1 is equal to
If x = f(t) cos t − f' (t) sin t and y = f(t) sin t + f'(t) cos t, then\[\left( \frac{dx}{dt} \right)^2 + \left( \frac{dy}{dt} \right)^2 =\]
If `x=a (cos t +t sint )and y= a(sint-cos t )` Prove that `Sec^3 t/(at),0<t< pi/2`
