Let $(M, g)$ be a Riemannian manifold. On one hand, we have a vector field $\textbf{X}\in \mathfrak{X}(M)$, where $\textbf{X}=\text{grad} W$ for some $W\in C^{\infty}(M)$. On the other hand, we the integral curves of $\text{grad} W$ are geodesic.
How could I prove that $$g(\text{grad} W,\text{grad} W)=\text{const}?$$
This assertion is found in the book "Differentiable Manifolds a Theoretical Thysics Approach" of Gerardo F. Torres del Castillo on page 140 equation (6.72).