What function will give as output the inclination measured in degrees of the tangent to the graph of $f$ at point $(x, f(x))$?

Question

I'm thinking of a function that would send back as output not the slope of the tangent (in the way the derivative function does) but the angle the tangent makes with the $x$-axis, for every point $(x, f(x))$ at which a tangent to the graph of $f$ exists.

I also would like the angle to be in degrees.

Certainly, some inverse trigonometric function is required here.

I once saw in a video by Herbert Gross (MIT, Calculus Revisited) that the derivative could be interpreted in a trigonometric fashion, something like $\frac{\text{opposite-side}}{\text{adjacent-side}}$)

Also, would this function be of any use?

Answer 1

If a line has slope $m$, then $\arctan(m)$ is the angle that line makes relative to the horizontal.

So you are seeking $\arctan(f'(x))$. And if you want it in degrees, then $\frac{180}{\pi}\arctan(f'(x))$.