Nocedal and Wright in their book Numerical Optimization have following expression in equation (3.5):
$\nabla f(x_k + \alpha_k p_k)^T p_k$
where $x_k$ is position, $\alpha_k$ is the step length, and $p_k$ is the line search direction at iterate $k$. Gradient is over the position.
They state this expression is simply the derivative $\phi'(\alpha_k)$, where $\phi(\alpha)=f(x_k+\alpha p_k)$ is a one dimensional function.
I can intuitively understand this statement, as the gradient times direction is the directional derivative, and derivative with respect to step length is right in that direction.
But, can you rigorously show these two are equivalent?