Dot product of the gradient of a function
Note that the derivative $d_pf$ of $f\colon\mathbb R^n\to\mathbb R$ in a point $p$ is not a vector, but a linear form instead.
In presence of an inner product $\langle.,.\rangle$ the gradient $\nabla^{\langle .,.\rangle}f(p)$ in respect to the inner product $\langle .,.\rangle$ is the unique vector which represents this linear form in presence of the specified inner product, that is $$d_pf(v)=\langle\nabla^{\langle .,.\rangle}f(p),v\rangle.$$