A curious property of simple functions

The other day, I was thinking about how, when you first learn calculus, the way you find the derivative of a function at a certain point is to find the slope of a line that passes through two very close points.

For example, for , we would find the derivative like this:

Once we know the derivative, we can easily calculate the tangent line at any given point . We'll call the function for "tangent line":

Those tangent lines neatly frame the original function, like so:

We can also find the lines that pass orthogonally through any point just as easily. We'll call that for "orthogonal line":

Which simplifies to:

Which in the case of comes out to:

If we draw a bunch of these orthogonal lines—the exact equivalent of the tagent lines but passing through at radians—we see an interesting V-shaped curve emerge.

I wanted to find the function that described this curve, so I started by find the point on that curve for any value of on the original function () by looking at the intersection of two orthogonal lines that were very close together:

Which we might otherwise state as:

That looks familiar: It's just the partial derivative of , , which in this case is:

which simplifies to:

or:

Plugging in into for and plotting a bunch of points does, in fact, recreate this curve:

We're almost there! Now we just need to substitute for in to get the orthogonal function:

Which simplifies to:

I don't have any idea if this has any application, but I find it surprising that a simple mental exercise in looking at orthogonal lines instead of tangent lines for such a basic function, , yields such a messy result for . And not every function has a valid orthogonal function. The only reason this worked is that we had a relationship between and (that is, ) that could be reversed to solve for , which isn't always the case. I'll write a future post at some point generalizing these orthogonal functions for any value of .