I encountered the following identities while reading this article on global calculus (p. 10):
$ d(\|df\|^2)=2\mathop{\iota_{\mathop{\mathrm{grad}} f}} \mathop{\mathrm{Hess}} f, $
$ \mathop{\mathrm{grad}}(\|df\|^2)=2\mathop{\nabla_{\mathop{\mathrm{grad}} f}}\mathop{\mathrm{grad}} f $
Here $\mathop{\mathrm{Hess}} f = \nabla d f$ is the covariant derivative of the 1-form $df$, and the norm $\|\cdot\|$ is given by Riemannian metric: $\|\upsilon\|^2=g(\upsilon,\upsilon)$.
I wonder how does one usually derive these identities (without using coordinates)?