Advertisements
Advertisements
प्रश्न
Differentiate\[\left( x + \frac{1}{x} \right)^x + x^\left( 1 + \frac{1}{x} \right)\] ?
Advertisements
उत्तर
\[\text{ Let y} = \left( x + \frac{1}{x} \right)^x + x^\left( 1 + \frac{1}{x} \right) \]
\[\text{ Also, Let u } = \left( x + \frac{1}{x} \right)^x \text{ and v } = x^\left( 1 + \frac{1}{x} \right) \]
\[ \therefore y = u + v\]
\[ \Rightarrow \frac{dy}{dx} = \frac{du}{dx} + \frac{dv}{dx} . . . \left( i \right)\]
\[\text{ Then, u } = \left( x + \frac{1}{x} \right)^x \]
\[ \Rightarrow \log u = \log \left( x + \frac{1}{x} \right)^x \]
\[ \Rightarrow \log u = x \log\left( x + \frac{1}{x} \right)\]
Differentiate both sides with respect to x,
\[\frac{1}{u}\frac{du}{dx} = \log\left( x + \frac{1}{x} \right)\frac{d}{dx}\left( x \right) + x\frac{d}{dx}\left[ \log\left( x + \frac{1}{x} \right) \right]\]
\[ \Rightarrow \frac{1}{u}\frac{du}{dx} = \log\left( x + \frac{1}{x} \right) + x\frac{1}{\left( x + \frac{1}{x} \right)}\frac{d}{dx}\left( x + \frac{1}{x} \right)\]
\[ \Rightarrow \frac{du}{dx} = u\left[ \log\left( x + \frac{1}{x} \right) + \frac{x}{\left( x + \frac{1}{x} \right)} \times \left( 1 - \frac{1}{x^2} \right) \right]\]
\[ \Rightarrow \frac{du}{dx} = \left( x + \frac{1}{x} \right)^x \left[ \log\left( x + \frac{1}{x} \right) + \frac{\left( x - \frac{1}{x} \right)}{\left( x + \frac{1}{x} \right)} \right]\]
\[ \Rightarrow \frac{du}{dx} = \left( x + \frac{1}{x} \right)^x \left[ \log\left( x + \frac{1}{x} \right) + \frac{x^2 - 1}{x^2 + 1} \right]\]
\[ \Rightarrow \frac{du}{dx} = \left( x + \frac{1}{x} \right)^x \left[ \frac{x^2 - 1}{x^2 + 1} + \log\left( x + \frac{1}{x} \right) \right]\]
\[\text{ Again }, v = x^\left( 1 + \frac{1}{x} \right) \]
\[ \Rightarrow \log v = \log\left[ x^\left( 1 + \frac{1}{x} \right) \right]\]
\[ \Rightarrow \log v = \left( 1 + \frac{1}{x} \right)\log x\]
Differentiating both sides with respect to x,
\[\frac{1}{v}\frac{dv}{dx} = \log x\frac{d}{dx}\left( 1 + \frac{1}{x} \right) + \left( 1 + \frac{1}{x} \right)\frac{d}{dx}\log x\]
\[ \Rightarrow \frac{1}{v}\frac{dv}{dx} = \left( - \frac{1}{x^2} \right)\log x + \left( 1 + \frac{1}{x} \right)\left( \frac{1}{x} \right)\]
\[ \Rightarrow \frac{1}{v}\frac{dv}{dx} = - \frac{\log x}{x^2} + \frac{1}{x} + \frac{1}{x^2}\]
\[ \Rightarrow \frac{dv}{dx} = v\left[ \frac{- \log x + x + 1}{x^2} \right]\]
\[ \Rightarrow \frac{dv}{dx} = x^\left( 1 + \frac{1}{x} \right) \left( \frac{x + 1 - \log x}{x^2} \right) . . . \left( iii \right)\]
\[\text{ From } \left( i \right), \left( ii \right) \text{and} \left( iii \right),\text{ we obtain }\]
\[\frac{dy}{dx} = \left( x + \frac{1}{x} \right)^x \left[ \frac{x^2 - 1}{x^2 + 1} + \log\left( x + \frac{1}{x} \right) \right] + x^\left( 1 + \frac{1}{x} \right) \left( \frac{x + 1 - log x}{x^2} \right)\]
APPEARS IN
संबंधित प्रश्न
If the sum of the lengths of the hypotenuse and a side of a right triangle is given, show that the area of the triangle is maximum, when the angle between them is 60º.
Differentiate the following functions from first principles eax+b.
Differentiate tan 5x° ?
Differentiate \[\sqrt{\frac{1 + x}{1 - x}}\] ?
Differentiate \[\frac{e^{2x} + e^{- 2x}}{e^{2x} - e^{- 2x}}\] ?
Differentiate \[\log \left( \tan^{- 1} x \right)\]?
Differentiate \[\log \sqrt{\frac{x - 1}{x + 1}}\] ?
If \[y = \sqrt{x^2 + a^2}\] prove that \[y\frac{dy}{dx} - x = 0\] ?
Differentiate \[\tan^{- 1} \left( \frac{\sqrt{1 + a^2 x^2} - 1}{ax} \right), x \neq 0\] ?
If \[y = \sin^{- 1} \left( \frac{2x}{1 + x^2} \right) + \sec^{- 1} \left( \frac{1 + x^2}{1 - x^2} \right), 0 < x < 1,\] prove that \[\frac{dy}{dx} = \frac{4}{1 + x^2}\] ?
Find \[\frac{dy}{dx}\] in the following case \[4x + 3y = \log \left( 4x - 3y \right)\] ?
If \[x y^2 = 1,\] prove that \[2\frac{dy}{dx} + y^3 = 0\] ?
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 \[y \sqrt{x^2 + 1} = \log \left( \sqrt{x^2 + 1} - x \right)\] ,Show that \[\left( x^2 + 1 \right) \frac{dy}{dx} + xy + 1 = 0\] ?
Differentiate \[\left( 1 + \cos x \right)^x\] ?
Differentiate \[x^\left( \sin x - \cos x \right) + \frac{x^2 - 1}{x^2 + 1}\] ?
Find \[\frac{dy}{dx}\] \[y = \frac{e^{ax} \cdot \sec x \cdot \log x}{\sqrt{1 - 2x}}\] ?
Find \[\frac{dy}{dx}\] \[y = x^{\sin x} + \left( \sin x \right)^x\] ?
Find \[\frac{dy}{dx}\]
\[y = x^x + x^{1/x}\] ?
If \[y = x \sin y\] , prove that \[\frac{dy}{dx} = \frac{y}{x \left( 1 - x \cos y \right)}\] ?
If \[y = \left( \sin x - \cos x \right)^{\sin x - \cos x} , \frac{\pi}{4} < x < \frac{3\pi}{4}, \text{ find} \frac{dy}{dx}\] ?
Find \[\frac{dy}{dx}\] , when \[x = \frac{3 at}{1 + t^2}, \text{ and } y = \frac{3 a t^2}{1 + t^2}\] ?
If \[x = 10 \left( t - \sin t \right), y = 12 \left( 1 - \cos t \right), \text { find } \frac{dy}{dx} .\] ?
Differentiate \[\sin^{- 1} \left( 2x \sqrt{1 - x^2} \right)\] with respect to \[\tan^{- 1} \left( \frac{x}{\sqrt{1 - x^2}} \right), \text { if }- \frac{1}{\sqrt{2}} < x < \frac{1}{\sqrt{2}}\] ?
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 \[\pi \leq x \leq 2\pi \text { and y } = \cos^{- 1} \left( \cos x \right), \text { find } \frac{dy}{dx}\] ?
If \[y = \sin^{- 1} \left( \frac{2x}{1 + x^2} \right)\] write the value of \[\frac{dy}{dx}\text { for } x > 1\] ?
If \[x^y = e^{x - y} ,\text{ then } \frac{dy}{dx}\] is __________ .
If \[\sin y = x \sin \left( a + y \right), \text { then }\frac{dy}{dx} \text { is}\] ____________ .
If \[f\left( x \right) = \sqrt{x^2 + 6x + 9}, \text { then } f'\left( x \right)\] is equal to ______________ .
If \[y = \frac{1}{1 + x^{a - b} +^{c - b}} + \frac{1}{1 + x^{b - c} + x^{a - c}} + \frac{1}{1 + x^{b - a} + x^{c - a}}\] then \[\frac{dy}{dx}\] is equal to ______________ .
Find the second order derivatives of the following function log (log x) ?
If y = x + tan x, show that \[\cos^2 x\frac{d^2 y}{d x^2} - 2y + 2x = 0\] ?
If \[y = e^{\tan^{- 1} x}\] prove that (1 + x2)y2 + (2x − 1)y1 = 0 ?
If y = sin (log x), prove that \[x^2 \frac{d^2 y}{d x^2} + x\frac{dy}{dx} + y = 0\] ?
If \[x = 3 \cos t - 2 \cos^3 t, y = 3\sin t - 2 \sin^3 t,\] find \[\frac{d^2 y}{d x^2} \] ?
If x = 2at, y = at2, where a is a constant, then find \[\frac{d^2 y}{d x^2} \text { at }x = \frac{1}{2}\] ?
\[\frac{d^{20}}{d x^{20}} \left( 2 \cos x \cos 3 x \right) =\]
Differentiate \[\tan^{- 1} \left( \frac{\sqrt{1 + x^2} - 1}{x} \right) w . r . t . \sin^{- 1} \frac{2x}{1 + x^2},\]tan-11+x2-1x w.r.t. sin-12x1+x2, if x ∈ (–1, 1) .
Find the height of a cylinder, which is open at the top, having a given surface area, greatest volume, and radius r.
