A step in the proof of Gauss's lemma in Riemannian geometry

Question

I am trying to understand the proof on Wikipedia of Gauss' lemma (which is more or less the same as in do Carno's textbook). I am having a hard time understanding the final step where in the notation of Wikipedia, we want to compute: $$\frac{\partial}{\partial s}\langle v+sw_n ,v+sw_n \rangle.$$ They state that this is equal to $2\langle v,w_n\rangle$ but I don't see why. Shouldn't the answer be $\frac{\partial}{\partial s}s^2||w_n|| = 2s||w_n||$? (note that $\langle v,w_n\rangle = 0$.)