Question

Find dy / dx for ex + y = y.

Answer 1

Answer: The required value of
`(dy)/(dx)` is
`(y)/(1-y).`

Step-by-step explanation: We are given to find the value of
`(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

`(d)/(dx)(x+y)=(d)/(dx)\ln y\\\\\\\Rightarrow 1+(dy)/(dx)=(1)/(y)(dy)/(dx)\\\\\\\Rightarrow \left((1)/(y)-1\right)(dy)/(dx)=1\\\\\\\Rightarrow (1-y)/(y)(dy)/(dx)=1\\\\\\\Rightarrow (dy)/(dx)=(y)/(1-y).`

Thus, the required value of
`(dy)/(dx)` is
`(y)/(1-y).`