Advertisements
Advertisements
प्रश्न
Differentiate \[x^{\cos^{- 1} x}\] ?
Advertisements
उत्तर
\[\text{ Let y } = x^{\cos^{- 1} x} . . . \left( i \right)\]
Taking log both sides
\[\log y = \log x^{\cos^{- 1} x} \]
\[ \Rightarrow \log y = \cos^{- 1} x \log x\]
Differentiating with respect to x,
\[\frac{1}{y}\frac{dy}{dx} = \cos^{- 1} x\frac{d}{dx}\left( \log x \right) + \log x\frac{d}{dx} \cos^{- 1} x \]
\[ \Rightarrow \frac{1}{y}\frac{dy}{dx} = \cos^{- 1} x\left( \frac{1}{x} \right) + \log x\left( \frac{- 1}{\sqrt{1 - x^2}} \right)\]
\[ \Rightarrow \frac{1}{y}\frac{dy}{dx} = \frac{\cos^{- 1} x}{x} - \frac{\log x}{\sqrt{1 - x^2}}\]
\[ \Rightarrow \frac{dy}{dx} = y\left[ \frac{\cos^{- 1} x}{x} - \frac{\log x}{\sqrt{1 - x^2}} \right]\]
\[ \Rightarrow \frac{dy}{dx} = x^{\cos^{- 1} x} \left[ \frac{\cos^{- 1} x}{x} - \frac{\log x}{\sqrt{1 - x^2}} \right] \left[ \text{ Using equation} \left( i \right) \right]\]
APPEARS IN
संबंधित प्रश्न
Differentiate \[3^{x^2 + 2x}\] ?
Differentiate \[\log \left( x + \sqrt{x^2 + 1} \right)\] ?
Differentiate \[3 e^{- 3x} \log \left( 1 + x \right)\] ?
Differentiate \[\frac{x^2 + 2}{\sqrt{\cos x}}\] ?
Differentiate \[\log \sqrt{\frac{x - 1}{x + 1}}\] ?
If \[y = e^x + e^{- x}\] prove that \[\frac{dy}{dx} = \sqrt{y^2 - 4}\] ?
Differentiate \[\tan^{- 1} \left( \frac{4x}{1 - 4 x^2} \right), - \frac{1}{2} < x < \frac{1}{2}\] ?
Differentiate \[\tan^{- 1} \left( \frac{x}{1 + 6 x^2} \right)\] ?
Differentiate \[\sin^{- 1} \left( \frac{1}{\sqrt{1 + x^2}} \right)\] with respect to x.
Find \[\frac{dy}{dx}\] in the following case \[xy = c^2\] ?
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 \[{10}^{ \log \sin x }\] ?
Find \[\frac{dy}{dx}\] \[y = x^{\cos x} + \left( \sin x \right)^{\tan x}\] ?
Find \[\frac{dy}{dx}\] \[y = x^x + \left( \sin x \right)^x\] ?
Find \[\frac{dy}{dx}\] \[y = x^{\log x }+ \left( \log x \right)^x\] ?
Find \[\frac{dy}{dx}\] , when \[x = b \sin^2 \theta \text{ and } y = a \cos^2 \theta\] ?
Find \[\frac{dy}{dx}\] , when \[x = \frac{1 - t^2}{1 + t^2} \text{ and y } = \frac{2 t}{1 + t^2}\] ?
Differentiate log (1 + x2) with respect to tan−1 x ?
Differentiate \[\sin^{- 1} \left( 2x \sqrt{1 - x^2} \right)\] with respect to \[\sec^{- 1} \left( \frac{1}{\sqrt{1 - x^2}} \right)\], if \[x \in \left( 0, \frac{1}{\sqrt{2}} \right)\] ?
If \[f\left( x \right) = \log \left\{ \frac{u \left( x \right)}{v \left( x \right)} \right\}, u \left( 1 \right) = v \left( 1 \right) \text{ and }u' \left( 1 \right) = v' \left( 1 \right) = 2\] , then find the value of `f' (1)` ?
The differential coefficient of f (log x) w.r.t. x, where f (x) = log x is ___________ .
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 \[\sin \left( x + y \right) = \log \left( x + y \right), \text { then } \frac{dy}{dx} =\] ___________ .
The derivative of \[\cos^{- 1} \left( 2 x^2 - 1 \right)\] with respect to \[\cos^{- 1} x\] is ___________ .
Find the second order derivatives of the following function x cos x ?
If y = ex cos x, prove that \[\frac{d^2 y}{d x^2} = 2 e^x \cos \left( x + \frac{\pi}{2} \right)\] ?
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 = sin (sin x), prove that \[\frac{d^2 y}{d x^2} + \tan x \cdot \frac{dy}{dx} + y \cos^2 x = 0\] ?
If y = ex (sin x + cos x) prove that \[\frac{d^2 y}{d x^2} - 2\frac{dy}{dx} + 2y = 0\] ?
If y log (1 + cos x), prove that \[\frac{d^3 y}{d x^3} + \frac{d^2 y}{d x^2} \cdot \frac{dy}{dx} = 0\] ?
If y = 3 e2x + 2 e3x, prove that \[\frac{d^2 y}{d x^2} - 5\frac{dy}{dx} + 6y = 0\] ?
If y = (cot−1 x)2, prove that y2(x2 + 1)2 + 2x (x2 + 1) y1 = 2 ?
\[\text { If x } = \cos t + \log \tan\frac{t}{2}, y = \sin t, \text { then find the value of } \frac{d^2 y}{d t^2} \text { and } \frac{d^2 y}{d x^2} \text { at } t = \frac{\pi}{4} \] ?
\[\frac{d^{20}}{d x^{20}} \left( 2 \cos x \cos 3 x \right) =\]
If x = f(t) and y = g(t), then \[\frac{d^2 y}{d x^2}\] is equal to
Differentiate the following with respect to x:
\[\cot^{- 1} \left( \frac{1 - x}{1 + x} \right)\]
