Question

If x = `e^(x/y)` then `dy/dx` = ______. 

Options

• `1-y/x`

• `1+y/x`

• `(x-y)/(xlogx)`

• `(x+y)/(xlogx)`

Answer 1

If x = `e^(x/y)` then `dy/dx` = `bbunderline((x-y)/(x  log  x))`. 

Explanation:

`x = e^(x/y)`

∴ `x/y` = log x    ...(1)

∴ y = `x/logx`

∴ `dy/dx = d/dx(x/logx)`

= `((logx).d/dx(x) - x.d/dx(logx))/((logx)`

= `((logx) xx 1 - x xx (1)/x)/((logx)^2`

= `(logx - 1)/((logx)(logx)`

= `(x/y - 1)/((x/y)(logx)`  ...[By (1)]

= `(x - y)/(xlogx)`