Say we have a function $F(\lambda) = \|f(\lambda)\|_{H^{-1}}$ where $\lambda \in \mathbb{R}$ and $H^{-1}(\mathbb{R}^n)$ is the usual dual of the Sobolev space $H^{1}_{0}(\mathbb{R}^n)$. Suppose we are interested in minimizing the function $F$, then is the following justified: $$\partial_{\lambda} F = \|\partial_{\lambda} f\|_{H^{-1}}$$ assuming $f$ smoothly depends on the parameter $\lambda?$

Edit [2]: Based on the comments, it does not make sense to take the norm of the derivative. How else might one proceed to minimize the $H^{-1}$ norm? As a toy example, let
$$f(\lambda) = \Delta u + \lambda u^p$$ for some $p>0, \lambda \in \mathbb{R}$ and $u\in H^{1}_0(\mathbb{R}^n).$ Then we can write, $$F(\lambda) = \|\Delta u + \lambda u^p\|_{H^{-1}}.$$ For what value of $\lambda$ does $F(\lambda)$ achieve it's minimum?