Question

if y =e*+y prove that dy/dx=y/1_y​

Answer 1

Answer:

The given expression is $y=e^{x+y}$

differentiate the above with respect to 'x'

$\frac{dy}{dx}=\frac{dy}{dx}e^{x+y}$

since, $\frac{dy}{dx}e^{x+y}=e^{x+y}(1+\frac{dy}{dx})$

$\frac{dy}{dx}=e^{x+y}(1+\frac{dy}{dx})$

$\frac{dy}{dx}=e^{x+y}+e^{x+y}\frac{dy}{dx}$

subtract both the sides by $e^{x+y}\frac{dy}{dx}$

$\frac{dy}{dx}-e^{x+y}\frac{dy}{dx}=e^{x+y}+e^{x+y}\frac{dy}{dx}$

$\frac{dy}{dx}(1-e^{x+y})=e^{x+y}$

since,  $y=e^{x+y}$

$\frac{dy}{dx}(1-y)=y$

divide both the sides by (1-y)

$\frac{dy}{dx}=\frac{y}{1-y}$

hence proved.