Advertisements
Advertisements
Question
Differentiate\[\left( x + \frac{1}{x} \right)^x + x^\left( 1 + \frac{1}{x} \right)\] ?
Solution
\[\text{ Let y} = \left( x + \frac{1}{x} \right)^x + x^\left( 1 + \frac{1}{x} \right) \]
\[\text{ Also, Let u } = \left( x + \frac{1}{x} \right)^x \text{ and v } = x^\left( 1 + \frac{1}{x} \right) \]
\[ \therefore y = u + v\]
\[ \Rightarrow \frac{dy}{dx} = \frac{du}{dx} + \frac{dv}{dx} . . . \left( i \right)\]
\[\text{ Then, u } = \left( x + \frac{1}{x} \right)^x \]
\[ \Rightarrow \log u = \log \left( x + \frac{1}{x} \right)^x \]
\[ \Rightarrow \log u = x \log\left( x + \frac{1}{x} \right)\]
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
RELATED QUESTIONS
Differentiate the following functions from first principles e−x.
Differentiate the following functions from first principles ecos x.
Differentiate \[e^{\sin} \sqrt{x}\] ?
Differentiate \[e^\sqrt{\cot x}\] ?
Differentiate \[e^x \log \sin 2x\] ?
If \[y = \frac{1}{2} \log \left( \frac{1 - \cos 2x }{1 + \cos 2x} \right)\] , prove that \[\frac{ dy }{ dx } = 2 \text{cosec }2x \] ?
Differentiate \[\cos^{- 1} \left\{ \frac{x}{\sqrt{x^2 + a^2}} \right\}\] ?
Differentiate \[\tan^{- 1} \left( \frac{4x}{1 - 4 x^2} \right), - \frac{1}{2} < x < \frac{1}{2}\] ?
Differentiate \[\sin^{- 1} \left( \frac{1 - x^2}{1 + x^2} \right) + \sec^{- 1} \left( \frac{1 + x^2}{1 - x^2} \right), x \in R\] ?
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 \[\frac{dy}{dx} = \frac{2}{1 + x^2} \] ?
Differentiate the following with respect to x:
\[\cos^{- 1} \left( \sin x \right)\]
Differentiate \[\sin^{- 1} \left\{ \frac{2^{x + 1} \cdot 3^x}{1 + \left( 36 \right)^x} \right\}\] with respect to x ?
Find \[\frac{dy}{dx}\] in the following case \[xy = c^2\] ?
Find \[\frac{dy}{dx}\] \[y = x^x + \left( \sin x \right)^x\] ?
If \[x^m y^n = 1\] , prove that \[\frac{dy}{dx} = - \frac{my}{nx}\] ?
If \[e^y = y^x ,\] prove that\[\frac{dy}{dx} = \frac{\left( \log y \right)^2}{\log y - 1}\] ?
Find \[\frac{dy}{dx}\] , when \[x = b \sin^2 \theta \text{ and } y = a \cos^2 \theta\] ?
Differentiate (log x)x with respect to log x ?
Differentiate\[\tan^{- 1} \left( \frac{\sqrt{1 + x^2} - 1}{x} \right)\] with respect to \[\sin^{-1} \left( \frac{2x}{1 + x^2} \right)\], If \[- 1 < x < 1, x \neq 0 .\] ?
Differentiate \[\sin^{- 1} \left( 2x \sqrt{1 - x^2} \right)\] with respect to \[\sec^{- 1} \left( \frac{1}{\sqrt{1 - x^2}} \right)\], if \[x \in \left( 0, \frac{1}{\sqrt{2}} \right)\] ?
Differentiate \[\tan^{- 1} \left( \frac{\cos x}{1 + \sin x} \right)\] with respect to \[\sec^{- 1} x\] ?
If \[f'\left( 1 \right) = 2 \text { and y } = f \left( \log_e x \right), \text { find} \frac{dy}{dx} \text { at }x = e\] ?
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 \[- \frac{\pi}{2} < x < 0 \text{ and y } = \tan^{- 1} \sqrt{\frac{1 - \cos 2x}{1 + \cos 2x}}, \text{ find } \frac{dy}{dx}\] ?
If \[\left| x \right| < 1 \text{ and y} = 1 + x + x^2 + . . \] to ∞, then find the value of \[\frac{dy}{dx}\] ?
The differential coefficient of f (log x) w.r.t. x, where f (x) = log x is ___________ .
If \[y = \sin^{- 1} \left( \frac{1 - x^2}{1 + x^2} \right), \text { then } \frac{dy}{dx} =\] _____________ .
If \[y = \sqrt{\sin x + y}, \text { then }\frac{dy}{dx} \text { equals }\] ______________ .
If \[y = \tan^{- 1} \left( \frac{\sin x + \cos x}{\cos x - \sin x} \right), \text { then } \frac{dy}{dx}\] is equal to ___________ .
Find the second order derivatives of the following function x3 + tan x ?
Find the second order derivatives of the following function tan−1 x ?
If y = e−x cos x, show that \[\frac{d^2 y}{d x^2} = 2 e^{- x} \sin x\] ?
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 log (1 + cos x), prove that \[\frac{d^3 y}{d x^3} + \frac{d^2 y}{d x^2} \cdot \frac{dy}{dx} = 0\] ?
\[\text { If x } = \cos t + \log \tan\frac{t}{2}, y = \sin t, \text { then find the value of } \frac{d^2 y}{d t^2} \text { and } \frac{d^2 y}{d x^2} \text { at } t = \frac{\pi}{4} \] ?
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 x = at2, y = 2 at, then \[\frac{d^2 y}{d x^2} =\]
If y = a cos (loge x) + b sin (loge x), then x2 y2 + xy1 =
f(x) = 3x2 + 6x + 8, x ∈ R
