My professor always writes the directional derivative with the condition that the functionis $C^1$ smooth.
i see him write many times the directional derivative written as $f'(x;d)=\nabla f^T d$, but right next to it he writes "because $\beta - C^1 smooth$", where $\beta$ is the Lipschitz constant, and $C^1$ means that it is continuously differentiable up to and including the first derivative.
Is the directional derivative $f'(x;d)$ ALWAYS equal to $\nabla f^T d$? or only under certain conditions?
EDIT And does "differentiable" mean with regards to a certain direction? Or if a function is differentiable, then does that mean it must be differentiable in all directions? If the former, then does $C^1$ mean it is differentiable in ALL directions?
That equality holds when $f$ is differentiable at $x$ (in the sense of having an appropriately good linear approximation at $x$). The standard theorem that one always proves is that $C^1$ implies differentiability. [And, of course, $(\nabla f(x))^\top$ will be the linear map that gives the good approximation.]