tan(x-y)=y/1+x^2

The reason for the derivations is a simple one. After an hour of playing with it, I still couldn’t figure out how to solve it (hats off to Alan).

A CDD related trick I learned in college was: any answer is better than none. An impressive looking answer was usually worth a few points. Over the long term, this bumped my score a half-letter grade, often bringing a D to a C. My grades often looked like CDD. That is better than DDD as some CDs have.

Wolfram Alpha says it is too hard so I am not going to think about it any more.

http://www.wolframalpha.com/input/?i=tan%28x-y%29%3Dy%2Bx^2

Perhaps you had intended some brackets???

I put this in Desmos as tan(x-y) = 1/(y + x^2).....it won't let me "capture" any  solutions, either...

Here's the graph...https://www.desmos.com/calculator/36ta7wen8n

Maybe you can make something of it.....good luck..!!!

Snarky comments: Anyone too dumb to use parenthetical operators in an equation is probably too dumb to understand this. However, there are others who will.

$$\; \text {Assuming}\\
\ tan(x-y) =\dfrac{y} {(1+x^2)}\; \Leftarrow \text {Find implicit derivative}\\
\ sec^2(x - y)(1 - y') = \dfrac{[(1+x^2)y'-2xy]}{(1+x^2)^2}\\
\ y' [\dfrac{ 1 } {(1 + x^2) + sec^2(x - y) ]} = \dfrac{2xy} {(1 + x^2)^2 + sec^2(x - y)}\\\\
\ y'[ 1+ x^2 + (1+x^2)^2 sec^2(x-y) ] = 2xy + (1+x^2)^2 sec^2(x-y)\\\\
\Large y'= \dfrac{2xy + (1+x^2)^2 sec^2(x-y)}{(1 + x^2) [ 1 + (1+x^2)sec^2(x-y)]} \\$$

Here's another.

$$\; \\
\ y^{\;'} =
\dfrac{-x^4\sec^2\left(x-y\right)-2x^2\sec ^2\left(x-y\right)-\sec^2\left(x-y\right)-2xy}{-x^4\sec ^2\left(x-y\right)-2x^2\sec ^2\left(x-y\right)-\sec ^2\left(x-y\right) -1 - x^2}$$

Two derivatives in the fountain which one should it be?

I am impressed with your derivation Nauseated.  I am only just starting to be comfortable with such derivatons, but why did you differentiate it in the first place?

I mean, the question is vague but nothing about finding a derivative is mentioned. 

Here's one way of looking at this:

I wrote that the solutions are real when |k|>=0.357....  I should have written that the solutions are real when |k|<=0.357... of course!

.

Thanks Alan, 

And yes, very amusing Nauseated