Let $f : S^{n-1} \subset \mathbb{R}^n \to \mathbb{R}^n \setminus \{0\}$ be a differentiable mapping, $n \geq 2$, and consider the function $F = \frac{f}{\|f\|} : S^{n-1} \to S^{n-1}$. I calculated the derivative of $F$ and arrived at
$$DF(p) \cdot v = \frac{1}{\|f(p)\|} \left( Df(p) \cdot v - \frac{\langle Df(p) \cdot v, f(p) \rangle}{\|f(p)\|^2} f(p) \right), \quad p \in S^{n-1}, \, v \in T_p S^{n-1}$$
Is it correct?