I am stuck trying to reach an (in)equality...
Let $\Omega \in \mathbb{R}$ and $f=f(t,x): \mathbb{R} \supset[0,T] \times \Omega \rightarrow \mathbb{R}$ be an element of
$$\mathcal{W}:=L_{_t}^2(0,T,H_{_x}^1(\Omega)):= \left\{ \varphi:[0,T]\times \Omega \rightarrow \mathbb{R} \, \Big| \, \int_{_{[0,T]}} \| \varphi (t,x) \|_{_{H_{_x}^1(\Omega)}}^{^2} dt < \infty \right\}.$$
I want to reach something like
$$ \frac{d}{dt}\Big\langle f(t,x),f(t,x) \Big\rangle_{_{\mathcal{W}}} \leq \Big\langle \dot f(t,x),f(t,x)\Big\rangle_{_{\mathcal{W}}}$$
to then extend and obtain
$$ \frac{d}{dt} \Big\|f(t,x) \Big\|_{_{\mathcal{W}}} = \frac{\tfrac{d}{dt}\Big\|f(t,x) \Big\|_{_{\mathcal{W}}}\Big\|f(t,x) \Big\|_{_{\mathcal{W}}}}{\Big\|f(t,x) \Big\|_{_{\mathcal{W}}}}\leq \frac{\Big\langle \dot f(t,x),f(t,x)\Big\rangle_{_{\mathcal{W}}}}{\Big\|f(t,x) \Big\|_{_{\mathcal{W}}}} $$
is this just straight forward? what am I missing?
For a real Hilbert space $H$, the square norm $N(h):=\langle h , h ⟩$ is (Fréchet) differentiable with derivative $dN(h)v = 2⟨ h, v ⟩ $. In this case we have $H=L^2_x$ where the subscript indicates that its the space of spatially $L^2$ functions. We then have the chain rule for functions $f=f(t,x)∈ H^1_tL^2_x = H^1([0,T],L^2(Ω)) = H^1(0,T;L^2(Ω))$, $$ \frac{d}{dt}‖f(t)‖_{L^2_x}^2 = \frac{d}{dt} (N \circ f)(t) = dN(f(t))f'(t) = 2⟨ f(t),f'(t)⟩ $$ At the same time, we have $\frac{d}{dt}‖f(t)‖_{L^2}^2 = 2‖f(t)‖_{L^2} \frac{d}{dt}‖f(t)‖_{L^2}$ by chain rule and the identity $(x^2)'= 2x$. Hence we have that $$\frac{d}{dt} ‖f(t)‖_{L^2} = \left\langle \frac{f(t)}{‖f(t)‖} , f'(t)\right\rangle$$
Note well that this is for functions in a different space than the one you prescribed $L^2_tH^1_x$,
$$L^2([0,T],H^1(Ω)) = \left\{ f:[0,T]\times Ω → \Bbb R \ \middle|\ \substack{ \displaystyle f(t),\partial_xf(t) ∈ L^2(dx) \text{ for a.e. }t\\ \displaystyle ∫_0^T ‖f‖^2_{L^2} + ‖\partial_x f‖^2_{L^2} \ dt< ∞} \right\}$$
The space we use is instead $H^1_xL^2_t = H^1([0,T],L^2(Ω))$, which is defined as $$H^1([0,T],L^2(Ω)) = \left\{ f:[0,T]\times Ω → \Bbb R \ \middle|\ \substack{ \displaystyle f(t),\partial_tf(t) ∈ L^2(dx) \text{ for a.e. $t$}\\ \displaystyle ∫_0^T ‖f‖^2_{L^2} + ‖\partial_t f‖^2_{L^2} \ dt< ∞} \right\}$$
(strictly speaking we don't need the top line in the definition since its implied by writing the second line.)