How to find the derivative of x + y2 = ln(x/y) ? show all workings and simplify! the answer is supposed to be y^2-x/xy

Step-by-step explanation:

If your function is

, that is definitely not the answer you should get after taking the derivative implicitely.  Rewrite your function to simplify a bit:

Take the derivative of x terms like "normal", but taking the derivative of y with respect to x has to be offset by dy/dx.  Doing that gives you:

Collect the terms with dy/dx on one side and everything else on the other side:

Now factor out the common dy/dx term, leaving this:

Now divide on the left to get dy/dx alone:

Simplify each set of fractions to get:

Bring the lower fraction up next to the top one and flip it upside down to multiply:

×

Simplifying that gives you the final result:

or you could multiply in the x on the bottom, as well.  Same difference as far as the solution goes.  You'd use this formula to find the slope of a function at a point by subbing in both the x and the y coordinates so it doesn't matter if you do the distribution at the very end or not.  You'll still get the same value for the slope.

0,0

step-by-step explanation: