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

Jun 11, 2016

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 $\frac{\mathrm{dy}}{\mathrm{dx}}$.

$\therefore \frac{d}{\mathrm{dx}} \ln \left(x - y\right) = \frac{d}{\mathrm{dx}} \left(x - y\right)$

$\therefore \frac{1}{x - y} \cdot \left(1 - \frac{\mathrm{dy}}{\mathrm{dx}}\right) = 1 - \frac{\mathrm{dy}}{\mathrm{dx}}$

$\therefore \frac{\mathrm{dy}}{\mathrm{dx}} \left(\frac{1}{x - y} - 1\right) = \left(\frac{1}{x - y} - 1\right)$

This seems to indicate the impossible fact of $\frac{\mathrm{dy}}{\mathrm{dx}} = 1$ for if this is true then $y = \int \mathrm{dx} = x$ and we cannot divide by zero, nor take the natural logarithm of zero.

Hence there is no solution.