Let's say that we have a function of one variable, $f\colon\Bbb{R}\to\Bbb{R}$, which is parametrized by a real parameter $\alpha$. For instance, let's say that we have the function $$ f(x;\alpha)=\exp(-\alpha x). $$ How can I compute the curvature of $f$ in each point $x\in\Bbb{R}$?
I am totally new to this field, so if you believe that the question must be close, please let me know. Any advice on the question will be much appreciated though.