Question

Find `("d"y)/("d"x)`, if xy = log(xy)

Answer 1

xy = log(xy)

Differentiating both sides w.r.t. x, we get

`x*("d"y)/("d"x) + y*"d"/("d"x)(x) = 1/(xy)*"d"/("d"x)(xy)`

∴ `x*("d"y)/("d"x) + y*1 = 1/(xy)[x*("d"y)/("d"x) + y*"d"/("d"x)(x)]`

∴ `x*("d"y)/("d"x) + y = 1/(xy)(x("d"y)/("d"x) + y*1)`

∴ `x*("d"y)/("d"x) + y = 1/y*("d"y)/("d"x) + 1/x`

∴ `(x - 1/y)("d"y)/("d"x) = 1/x - y`

∴ `-((1 - xy)/y)("d"y)/("d"x) = ((1 - xy)/x)`

∴ `("d"y)/("d"x) = -((1 - xy)/x) xx (y/(1 - xy))`

∴ `("d"y)/("d"x) = (-y)/x`