Question

Find dy / dx for ex + y = y.

Answer 1

Answer:  The required value of $\dfrac{dy}{dx}$ is $\dfrac{y}{1-y}.$

Step-by-step explanation:   We are given to find the value of $\dfrac{dy}{dx}$ from the following equation :

$e^{x+y}=y~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(i)$

Taking natural logarithm on both sides of equation (i), we have

$\ln e^{x+y}=\ln y\\\\\Rightarrow x+y=\ln y~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(ii)$

Differentiating both sides of equation (ii) with respect to x, we get

$\dfrac{d}{dx}(x+y)=\dfrac{d}{dx}\ln y\\\\\\\Rightarrow 1+\dfrac{dy}{dx}=\dfrac{1}{y}\dfrac{dy}{dx}\\\\\\\Rightarrow \left(\dfrac{1}{y}-1\right)\dfrac{dy}{dx}=1\\\\\\\Rightarrow \dfrac{1-y}{y}\dfrac{dy}{dx}=1\\\\\\\Rightarrow \dfrac{dy}{dx}=\dfrac{y}{1-y}.$

Thus, the required value of $\dfrac{dy}{dx}$ is $\dfrac{y}{1-y}.$