Advertisements
Advertisements
Question
If \[e^{x + y} - x = 0\] ,prove that \[\frac{dy}{dx} = \frac{1 - x}{x}\] ?
Advertisements
Solution
\[\text{ We have}, e^{x + y} - x = 0\]
\[ \Rightarrow e^{x + y} = x . . . \left( 1 \right)\]
Differentiating with respect to x using chain rule,
\[\frac{d}{dx}\left( e^{x + y} \right) = \frac{d}{dx}\left( x \right)\]
\[ \Rightarrow e^{x + y} \frac{d}{dx}\left( x + y \right) = 1\]
\[ \Rightarrow x\left[ 1 + \frac{dy}{dx} \right] = 1 \left[ \text{ Using equation } \left( i \right) \right]\]
\[ \Rightarrow 1 + \frac{dy}{dx} = \frac{1}{x}\]
\[ \Rightarrow \frac{dy}{dx} = \frac{1}{x} - 1\]
\[ \Rightarrow \frac{dy}{dx} = \frac{1 - x}{x}\]
APPEARS IN
RELATED QUESTIONS
Show that the semi-vertical angle of the cone of the maximum volume and of given slant height is `cos^(-1)(1/sqrt3)`
Differentiate \[3^{x \log x}\] ?
Differentiate \[\frac{e^x \log x}{x^2}\] ?
Differentiate \[\frac{3 x^2 \sin x}{\sqrt{7 - x^2}}\] ?
Differentiate \[e^x \log \sin 2x\] ?
Differentiate \[\cos^{- 1} \left\{ \frac{\cos x + \sin x}{\sqrt{2}} \right\}, - \frac{\pi}{4} < x < \frac{\pi}{4}\] ?
Differentiate \[\tan^{- 1} \left\{ \frac{x}{1 + \sqrt{1 - x^2}} \right\}, - 1 < x < 1\] ?
Differentiate \[\cos^{- 1} \left( \frac{1 - x^{2n}}{1 + x^{2n}} \right), < x < \infty\] ?
Find \[\frac{dy}{dx}\] in the following case \[x^{2/3} + y^{2/3} = a^{2/3}\] ?
If \[\sqrt{1 - x^2} + \sqrt{1 - y^2} = a \left( x - y \right)\] , prove that \[\frac{dy}{dx} = \frac{\sqrt{1 - y^2}}{1 - x^2}\] ?
If \[\log \sqrt{x^2 + y^2} = \tan^{- 1} \left( \frac{y}{x} \right)\] Prove that \[\frac{dy}{dx} = \frac{x + y}{x - y}\] ?
Differentiate \[x^{x \cos x +} \frac{x^2 + 1}{x^2 - 1}\] ?
If \[x^y + y^x = \left( x + y \right)^{x + y} , \text{ find } \frac{dy}{dx}\] ?
If \[e^x + e^y = e^{x + y}\] , prove that
\[\frac{dy}{dx} + e^{y - x} = 0\] ?
If \[y = \sqrt{\cos x + \sqrt{\cos x + \sqrt{\cos x + . . . to \infty}}}\] , prove that \[\frac{dy}{dx} = \frac{\sin x}{1 - 2 y}\] ?
Find \[\frac{dy}{dx}\] , when \[x = b \sin^2 \theta \text{ and } y = a \cos^2 \theta\] ?
Find \[\frac{dy}{dx}\] ,when \[x = \frac{e^t + e^{- t}}{2} \text{ and } y = \frac{e^t - e^{- t}}{2}\] ?
Find \[\frac{dy}{dx}\] , when \[x = \frac{1 - t^2}{1 + t^2} \text{ and y } = \frac{2 t}{1 + t^2}\] ?
If \[x = 2 \cos \theta - \cos 2 \theta \text{ and y} = 2 \sin \theta - \sin 2 \theta\], prove that \[\frac{dy}{dx} = \tan \left( \frac{3 \theta}{2} \right)\] ?
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 \[x = \left( t + \frac{1}{t} \right)^a , y = a^{t + \frac{1}{t}} , \text{ find } \frac{dy}{dx}\] ?
Differentiate log (1 + x2) with respect to tan−1 x ?
Differentiate \[\sin^{- 1} \sqrt{1 - x^2}\] with respect to \[\cos^{- 1} x, \text { if}\]\[x \in \left( 0, 1 \right)\] ?
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 \[\sin^{- 1} \left( 2x \sqrt{1 - x^2} \right)\] 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 \[y = \log \sqrt{\tan x}, \text{ write } \frac{dy}{dx} \] ?
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 derivative of the function \[\cot^{- 1} \left| \left( \cos 2 x \right)^{1/2} \right| \text{ at } x = \pi/6 \text{ is }\] ______ .
If y = x + tan x, show that \[\cos^2 x\frac{d^2 y}{d x^2} - 2y + 2x = 0\] ?
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 x = 4z2 + 5, y = 6z2 + 7z + 3, find \[\frac{d^2 y}{d x^2}\] ?
\[\text { If x } = a\left( \cos t + t \sin t \right) \text { and y} = a\left( \sin t - t \cos t \right),\text { then find the value of } \frac{d^2 y}{d x^2} \text { at } t = \frac{\pi}{4} \] ?
If y = |x − x2|, then find \[\frac{d^2 y}{d x^2}\] ?
If \[y = \left| \log_e x \right|\] find\[\frac{d^2 y}{d x^2}\] ?
If x = 2 at, y = at2, where a is a constant, then \[\frac{d^2 y}{d x^2} \text { at x } = \frac{1}{2}\] is
If \[y = \frac{ax + b}{x^2 + c}\] then (2xy1 + y)y3 =
If \[y = \log_e \left( \frac{x}{a + bx} \right)^x\] then x3 y2 =
