If A is a linear operator defined on $\mathbb{L} ^2 _\mathbb{R}$ of $\frac{d}{dx}$ then its adjoint must satisfy the property: $$(A^*f,g) = (f,Ag) \\ f,g \in \mathbb{L} ^2 _\mathbb{R}$$$
Now if $A = \frac{d}{dx}$, then how can I find $A^*$? I am unsure of this because first how is the scalar product defined on $\mathbb{L} ^2 _\mathbb{R}$? Do I just need to "plug in" $f,g$ in the scalar product and then work it out from there? I know the answer is $-\frac{d}{dx}$.