Question

If 2^x + 2^y = 2^x+y, then `(dy)/(dx)` is equal to ______.

Options

• `((2^x + 2^y))/((2^x - 2^y))`

• `((2^x + 2^y))/((1 + 2^(x+y))`

• `2^(x-y).(2^y - 1)/(1 - 2^x)`

• `((2^(x + y) - 2^x))/2^y`

Answer 1

If 2^x + 2^y = 2^x+y, then `(dy)/(dx)` is equal to `underlinebb(2^(x-y).(2^y - 1)/(1 - 2^x))`.

Explanation:

2^x + 2^y = 2^x+y

⇒ `2^x ln2 + 2^y ln2 (dy)/(dx) = 2^(x + y) ln2(1 + (dy)/(dx))`

`(dy)/(dx) = ((2^x ln2 - 2^y2^x ln2))/((-2^y  ln2 + 2^y2^x ln2))`

= `-2^(x - y)[(1 - 2^y)/(1 - 2^x)]`

`(dy)/(dx) = 2^(x-y)[(2^y - 1)/(1 - 2^x)]`