Angle between function and a horizontal line

Question

How do I find the angle between a function of $x$ and the $x$-axis given the following information:

$$y = 15 \ln\frac{x}{80}$$ When there is an $x$-intercept at $x=80$?

I'm pretty sure I need to take the derivative of $y(x)$ to find the tangent line, but I'm not sure where to go after that.

Thank you

Answer 1

The derivative gives you the the line tangent to the function at the point x. If you get like y = ax + b you can get the angle with the tangent function, since the tangent of the angle gives you the inclination of the line

Answer 2

The derivative $y^\prime(80)$ is the slope of the curve there (at the $x$-intercept). The angle is the inverse tangent of that.

$$\theta = \arctan y^\prime(80).$$

Answer 3

The value of the derivative is the slope of the tangent line, and the slope of a line is the tangent of the angle the line makes with respect to the $x$ axis. (Note the two different uses of the word "tangent.")