I need to show that the linear operator
$ \frac{d}{dx}: \mathcal{D} \rightarrow L^2(-\infty,\infty)$ is bounded with respect to the inner product
$(f,g) = \int_{-\infty}^{\infty} {f(t) \overline{g(t)}+f'(t)\overline{g'(t)}} dt $
on $\mathcal{D}$.
Here, $\mathcal{D}$ represents the space of differentiable functions $f$, where $f, f' \in L^2(-\infty,\infty)$.
My work
To prove the boundedness, I work on $||\frac{d}{dx}f||^2$ which is equal to $||f'||_2^2 +||f''||_2^2$ which takes me to the norm of the second derivative. So, I tried to rewrite it in terms of norm of $f$ and $f'$ using integration by parts, but this does not seems work as parts gives integrals with third derivative terms.
It seems like you want to give that inner product to $\mathcal{D}$. This makes sense, since this is the inner product on the Sobolev space $H^1$ (the inner product doesn't make sense on all of $L^2$, as not every function in $L^2$ has a weak derivative in $L^2$), and $\frac{d}{dx}:H^1\rightarrow L^2$ boundedly. Indeed, $$\left\|\frac{d}{dx}f\right\|_{L^2}\leq \|f\|_{L^2}+\left\|\frac{d}{dx}f\right\|_{L^2}\leq \sqrt{2}\|f\|_{H^1},$$ which proves boundedness.