Consider the system $\dot x = f(x)$ with $f(0)=0$, f continuously differentiable and $\frac{\partial f}{\partial x}(x)^TP+P\frac{\partial f}{\partial x}(x)\preceq-I$ for all x and some $P=P^T\succ0$.
How to verify that f(x) can be written as $f(x)=\int_0^1\frac{\partial f}{\partial x}(\sigma x)xd\sigma$, and use this to show that $\forall x:f(x)^TPx+x^TPf(x)\leq-x^Tx$?