Question

How do you implicitly differentiate `ln(x-y)=x-y`?

Answer 1

It appears as if there is no solution.

Explanation:

Since we cannot get y explicitly as a function of x, we use implicit differentiation.

Differentiate both sides, bearing in mind that y is a function of x, and then solve for `dy/dx`.

`therefore d/dxln(x-y)=d/dx(x-y)`

`therefore1/(x-y)*(1-(dy)/dx)=1-(dy)/dx`

`therefore(dy)/dx(1/(x-y)-1)=(1/(x-y)-1)`

This seems to indicate the impossible fact of `(dy)/dx=1` for if this is true then `y=intdx=x` and we cannot divide by zero, nor take the natural logarithm of zero.

Hence there is no solution.