Advertisements
Advertisements
Question
If \[y^x + x^y + x^x = a^b\] ,find \[\frac{dy}{dx}\] ?
Solution
\[\text{ Given that }y^x + x^y + x^x = a^b \]
\[\text{ Putting u }= y^x , v = x^y \text{and }w = x^x , \text{ we get }\]
\[ u + v + w = a^b \]
\[ \therefore \frac{du}{dx} + \frac{dv}{dx} + \frac{dw}{dx} = 0 . . . \left( i \right)\]
\[\text{ Now, u } = y^x \]
Taking log on both sides,
\[\log u = x \log y\]
\[\Rightarrow \frac{1}{u}\frac{du}{dx} = x\frac{d}{dx}\left( \log y \right) + \log y\frac{d}{dx}\left( x \right) \left[ \text{ using product } rule \right]\]
\[ \Rightarrow \frac{1}{u}\frac{du}{dx} = x\frac{1}{y}\frac{dy}{dx} + \log y \times 1\]
\[ \Rightarrow \frac{du}{dx} = u\left( \frac{x}{y}\frac{dy}{dx} + \log y \right)\]
\[ \Rightarrow \frac{du}{dx} = y^x \left( \frac{x}{y}\frac{dy}{dx} + \log y \right) . . . \left( ii \right)\]
\[\text{ Also, v } = x^y\]
Taking log on both sides,
\[\log v = y \log x\]
\[\Rightarrow \frac{1}{v}\frac{dv}{dx} = y\frac{d}{dx}\left( \log x \right) + \log x\frac{dy}{dx}\]
\[ \Rightarrow \frac{1}{v}\frac{dv}{dx} = y\frac{1}{x} + \log x\frac{dy}{dx}\]
\[ \Rightarrow \frac{dv}{dx} = v\left[ \frac{y}{x} + \log x\frac{dy}{dx} \right]\]
\[ \Rightarrow \frac{dv}{dx} = x^y \left[ \frac{y}{x} + \log x\frac{dy}{dx} \right] . . . \left( iii \right)\]
\[\text{ Again, w } = x^x\]
Taking log on both sides,
\[\log w = x \log x\]
\[\Rightarrow \frac{1}{w}\frac{dw}{dx} = x\frac{d}{dx}\left( \log x \right) + \log x\frac{d}{dx}\left( x \right)\]
\[ \Rightarrow \frac{1}{w}\frac{dw}{dx} = x\frac{1}{x} + \log x\left( 1 \right)\]
\[ \Rightarrow \frac{dw}{dx} = w\left( 1 + \log x \right)\]
\[ \Rightarrow \frac{dw}{dx} = x^x \left( 1 + \log x \right) . . . \left( iv \right)\]
\[\text{ From } \left( i \right), \left( ii \right), \left( iii \right)\text{ and }\left( iv \right), \text{ we have }\]
\[ y^x \left( \frac{x}{y}\frac{dy}{dx} + \log y \right) + x^y \left( \frac{y}{x} + \log x\frac{dy}{dx} \right) + x^x \left( 1 + \log x \right) = 0\]
\[ \Rightarrow \left( x . y^{x - 1} + x^y . \log x \right)\frac{dy}{dx} = - x^x \left( 1 + \log x \right) - y . x^{y - 1} - y^x \log y\]
\[ \therefore \frac{dy}{dx} = \frac{- \left\{ y^x \log y + y . x^{y - 1} + x^x \left( 1 + \log x \right) \right\}}{x . y^{x - 1} + x^y \log x}\]
APPEARS IN
RELATED QUESTIONS
Differentiate tan2 x ?
Differentiate `2^(x^3)` ?
Differentiate \[\sin \left( \frac{1 + x^2}{1 - x^2} \right)\] ?
Differentiate \[e^{\tan^{- 1}} \sqrt{x}\] ?
Differentiate \[\cos^{- 1} \left\{ 2x\sqrt{1 - x^2} \right\}, \frac{1}{\sqrt{2}} < x < 1\] ?
Differentiate \[\cos^{- 1} \left( \frac{x + \sqrt{1 - x^2}}{\sqrt{2}} \right), - 1 < x < 1\] ?
Differentiate \[\tan^{- 1} \left( \frac{a + x}{1 - ax} \right)\] ?
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 \[\frac{dy}{dx} = \frac{4}{1 + x^2}\] ?
Differentiate \[\sin^{- 1} \left\{ \frac{2^{x + 1} \cdot 3^x}{1 + \left( 36 \right)^x} \right\}\] with respect to x ?
If \[x \sin \left( a + y \right) + \sin a \cos \left( a + y \right) = 0\] Prove that \[\frac{dy}{dx} = \frac{\sin^2 \left( a + y \right)}{\sin a}\] ?
If \[\sin \left( xy \right) + \frac{y}{x} = x^2 - y^2 , \text{ find} \frac{dy}{dx}\] ?
Differentiate \[x^{1/x}\] with respect to x.
Differentiate \[\left( \log x \right)^x\] ?
Differentiate \[\left( \log x \right)^{ \log x }\] ?
Differentiate \[\left( x \cos x \right)^x + \left( x \sin x \right)^{1/x}\] ?
If \[x^y + y^x = \left( x + y \right)^{x + y} , \text{ find } \frac{dy}{dx}\] ?
If \[x \sin \left( a + y \right) + \sin a \cos \left( a + y \right) = 0\] , prove that \[\frac{dy}{dx} = \frac{\sin^2 \left( a + y \right)}{\sin a}\] ?
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 \[xy = e^{x - y} , \text{ find } \frac{dy}{dx}\] ?
If \[y = \sqrt{x + \sqrt{x + \sqrt{x + . . . to \infty ,}}}\] prove that \[\frac{dy}{dx} = \frac{1}{2 y - 1}\] ?
Find \[\frac{dy}{dx}\] , when \[x = \cos^{- 1} \frac{1}{\sqrt{1 + t^2}} \text{ and y } = \sin^{- 1} \frac{t}{\sqrt{1 + t^2}}, t \in R\] ?
If \[x = \frac{\sin^3 t}{\sqrt{\cos 2 t}}, y = \frac{\cos^3 t}{\sqrt{\cos t 2 t}}\] , find\[\frac{dy}{dx}\] ?
If \[x = a \left( \frac{1 + t^2}{1 - t^2} \right) \text { and y } = \frac{2t}{1 - t^2}, \text { find } \frac{dy}{dx}\] ?
\[\text { If }x = \cos t\left( 3 - 2 \cos^2 t \right), y = \sin t\left( 3 - 2 \sin^2 t \right) \text { find the value of } \frac{dy}{dx}\text{ at }t = \frac{\pi}{4}\] ?
Write the derivative of sinx with respect to cos x ?
If \[y = \sin^{- 1} \left( \frac{2x}{1 + x^2} \right)\] write the value of \[\frac{dy}{dx}\text { for } x > 1\] ?
If \[y = \tan^{- 1} \left( \frac{1 - x}{1 + x} \right), \text{ find} \frac{dy}{dx}\] ?
If \[y = \sin^{- 1} \left( \frac{1 - x^2}{1 + x^2} \right) + \cos^{- 1} \left( \frac{1 - x^2}{1 + x^2} \right),\text{ find } \frac{dy}{dx}\] ?
If \[x = 3\sin t - \sin3t, y = 3\cos t - \cos3t \text{ find }\frac{dy}{dx} \text{ at } t = \frac{\pi}{3}\] ?
The differential coefficient of f (log x) w.r.t. x, where f (x) = log x is ___________ .
If \[f\left( x \right) = \left( \frac{x^l}{x^m} \right)^{l + m} \left( \frac{x^m}{x^n} \right)^{m + n} \left( \frac{x^n}{x^l} \right)^{n + 1}\] the f' (x) is equal to _____________ .
If y = e−x cos x, show that \[\frac{d^2 y}{d x^2} = 2 e^{- x} \sin x\] ?
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 \[y = e^{\tan^{- 1} x}\] prove that (1 + x2)y2 + (2x − 1)y1 = 0 ?
If y = cot x show that \[\frac{d^2 y}{d x^2} + 2y\frac{dy}{dx} = 0\] ?
If y = ex (sin x + cos x) prove that \[\frac{d^2 y}{d x^2} - 2\frac{dy}{dx} + 2y = 0\] ?
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 f(x) = (cos x + i sin x) (cos 2x + i sin 2x) (cos 3x + i sin 3x) ...... (cos nx + i sin nx) and f(1) = 1, then f'' (1) is equal to
If xy = e(x – y), then show that `dy/dx = (y(x-1))/(x(y+1)) .`
f(x) = xx has a stationary point at ______.
