Understanding the difference between $\partial f, \mathcal{D}_{f^{*}}, f^{*}, \partial f^{*}$ for a specific set of functions I am trying to calculate the following quantities: $\partial f, \mathcal{D}_{f^{*}}, f^{*}, \partial f^{*}$ for the following functions:

*

*$f(x) = x^2, \mathcal{D}_{f} = \mathbb{R}$.

*$f(x) = |x|, \mathcal{D}_{f} = \mathbb{R}$.

*$f(x) = e^x, \mathcal{D}_{f} = \mathbb{R}$.

*$f(x) = \dfrac{|x|^{p}}{p}, \mathcal{D}_{f} = \mathbb{R}, p>1$.

*$f(x) = x \log x-x+1, f(0) = 1, \mathcal{D}_{f} = [0, + \infty]$.

I am completely lost, seeing that as far as my study goes I have not been taught how to "calculate" them, other than the relatively well known definition.
Any pointers would be more than welcome.

Any book with Convex Analysis in the title will help you. The function in 2 is sublinear so its conjugate must be an indicator function. I find the following formula very useful for computing the subdifferential:
$$\partial f(x) = \mathbb{R} \cap [f'_-(x),f'_+(x)],$$
where $f'_{\pm}$ are the one-sided derivatives from Calculus I.
