I know the prescribed method is to define the function $g(t) = f(\mathbf{x}^k - t\nabla f(\mathbf{x}^k))$ and then to differentiate this and set = 0, however calculating $g'(t)$ is somewhat confusing for me.
$$\frac{d}{dt}[g(t)] = \frac{d}{dt}[f(\mathbf{x}^k - t\nabla f(\mathbf{x}^k))] = -\nabla f(\mathbf{x}^k)^T * \frac{df}{dt}((\mathbf{x}^k - t\nabla f(\mathbf{x}^k)))$$
My books say the expression is equivalent to $-\nabla f(\mathbf{x}^k)^T \nabla f(\mathbf{x}^k - t\nabla f(\mathbf{x}^k))$. I do not understand how this is a derivative w.r.t to $t$.