Advertisements
Advertisements
प्रश्न
If \[x^x + y^x = 1\], prove that \[\frac{dy}{dx} = - \left\{ \frac{x^x \left( 1 + \log x \right) + y^x \cdot \log y}{x \cdot y^\left( x - 1 \right)} \right\}\] ?
Advertisements
उत्तर
\[\text{ We have}, x^x + y^x = 1\]
\[ \Rightarrow e^{\log x^x} + e^{\log y^x} = 1\]
\[ \Rightarrow e^{x \log x} + e^{x \log y} = 1 \]
Differentiating with respect to x using chain rule,
\[\frac{d}{dx}\left( e^{x\log x} \right) + \frac{d}{dx}\left( e^{x \log y} \right) = \frac{d}{dx}\left( 1 \right)\]
\[ \Rightarrow e^{x \log x} \frac{d}{dx}\left( x \log x \right) + e^{x \log y} \frac{d}{dx}\left( x \log y \right) = 0\]
\[ \Rightarrow e^{x \log x} \left[ x\frac{d}{dx}\left( \log x \right) + \log x\frac{d}{dx}\left( x \right) \right] + e^{\log y^x} \left[ x\frac{d}{dx}\left( \log y \right) + \log y\frac{d}{dx}\left( x \right) \right] = 0\]
\[ \Rightarrow x^x \left[ x\left( \frac{1}{x} \right) + \log x\left( 1 \right) \right] + y^x \left[ x\left( \frac{1}{y} \right)\frac{dy}{dx} + \log y\left( 1 \right) \right] = 0\]
\[ \Rightarrow x^x \left[ 1 + \log x \right] + y^x \left( \frac{x}{y}\frac{dy}{dx} + \log y \right) = 0\]
\[ \Rightarrow y^x \times \frac{x}{y}\frac{dy}{dx} = - \left[ x^x \left( 1 + \log x \right) + y^x \log y \right]\]
\[ \Rightarrow \left( x y^{x - 1} \right)\frac{dy}{dx} = - \left[ x^x \left( 1 + \log x \right) + y^x \log y \right]\]
\[ \Rightarrow \frac{dy}{dx} = - \left[ \frac{x^x \left( 1 + \log x \right) + y^x \log y}{x y^{x - 1}} \right]\]
APPEARS IN
संबंधित प्रश्न
Differentiate \[3^{x^2 + 2x}\] ?
Differentiate \[\sin \left( \frac{1 + x^2}{1 - x^2} \right)\] ?
Differentiate \[\tan \left( e^{\sin x }\right)\] ?
Differentiate \[\log \left( x + \sqrt{x^2 + 1} \right)\] ?
Differentiate \[e^{\sin^{- 1} 2x}\] ?
Differentiate \[\sin^2 \left\{ \log \left( 2x + 3 \right) \right\}\] ?
Differentiate \[\log \left( \cos x^2 \right)\] ?
If \[y = \frac{e^x - e^{- x}}{e^x + e^{- x}}\] .prove that \[\frac{dy}{dx} = 1 - y^2\] ?
Differentiate \[\cos^{- 1} \left( \frac{1 - x^{2n}}{1 + x^{2n}} \right), < x < \infty\] ?
Differentiate \[\tan^{- 1} \left( \frac{\sqrt{x} + \sqrt{a}}{1 - \sqrt{xa}} \right)\] ?
Find \[\frac{dy}{dx}\] in the following case \[\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\] ?
If \[x y^2 = 1,\] prove that \[2\frac{dy}{dx} + y^3 = 0\] ?
If \[e^x + e^y = e^{x + y} , \text{ prove that } \frac{dy}{dx} = - \frac{e^x \left( e^y - 1 \right)}{e^y \left( e^x - 1 \right)} or \frac{dy}{dx} + e^{y - x} = 0\] ?
If \[\cos y = x \cos \left( a + y \right), \text{ with } \cos a \neq \pm 1, \text{ prove that } \frac{dy}{dx} = \frac{\cos^2 \left( a + y \right)}{\sin a}\] ?
Find \[\frac{dy}{dx}\] \[y = x^x + \left( \sin x \right)^x\] ?
If \[x^y + y^x = \left( x + y \right)^{x + y} , \text{ find } \frac{dy}{dx}\] ?
If \[xy \log \left( x + y \right) = 1\] , prove that \[\frac{dy}{dx} = - \frac{y \left( x^2 y + x + y \right)}{x \left( x y^2 + x + y \right)}\] ?
Find the derivative of the function f (x) given by \[f\left( x \right) = \left( 1 + x \right) \left( 1 + x^2 \right) \left( 1 + x^4 \right) \left( 1 + x^8 \right)\] and hence find `f' (1)` ?
If \[\frac{dy}{dx}\] when \[x = a \cos \theta \text{ and } y = b \sin \theta\] ?
Find \[\frac{dy}{dx}\], when \[x = a \left( \cos \theta + \theta \sin \theta \right) \text{ and }y = a \left( \sin \theta - \theta \cos \theta \right)\] ?
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 \[x = \frac{1 + \log t}{t^2}, y = \frac{3 + 2\log t}{t}, \text { find } \frac{dy}{dx}\] ?
Differentiate (log x)x with respect to log x ?
Differentiate \[\tan^{- 1} \left( \frac{2x}{1 - x^2} \right)\] with respect to \[\cos^{- 1} \left( \frac{1 - x^2}{1 + x^2} \right),\text { if }0 < x < 1\] ?
Differentiate \[\sin^{- 1} \left( 2 ax \sqrt{1 - a^2 x^2} \right)\] with respect to \[\sqrt{1 - a^2 x^2}, \text{ if }-\frac{1}{\sqrt{2}} < ax < \frac{1}{\sqrt{2}}\] ?
If f (x) = loge (loge x), then write the value of `f' (e)` ?
If \[\left| x \right| < 1 \text{ and y} = 1 + x + x^2 + . . \] to ∞, then find the value of \[\frac{dy}{dx}\] ?
If \[f\left( x \right) = \sqrt{x^2 + 6x + 9}, \text { then } f'\left( x \right)\] is equal to ______________ .
Find the second order derivatives of the following function ex sin 5x ?
Find the second order derivatives of the following function tan−1 x ?
If \[y = \frac{\log x}{x}\] show that \[\frac{d^2 y}{d x^2} = \frac{2 \log x - 3}{x^3}\] ?
If x = a (θ − sin θ), y = a (1 + cos θ) prove that, find \[\frac{d^2 y}{d x^2}\] ?
If x = sin t, y = sin pt, prove that \[\left( 1 - x^2 \right)\frac{d^2 y}{d x^2} - x\frac{dy}{dx} + p^2 y = 0\] ?
\[\text { If }y = A e^{- kt} \cos\left( pt + c \right), \text { prove that } \frac{d^2 y}{d t^2} + 2k\frac{d y}{d t} + n^2 y = 0, \text { where } n^2 = p^2 + k^2 \] ?
If y = a cos (loge x) + b sin (loge x), then x2 y2 + xy1 =
If y = xn−1 log x then x2 y2 + (3 − 2n) xy1 is equal to
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) .
f(x) = 3x2 + 6x + 8, x ∈ R
