Normal unit vector of shape operator

Question

Let $\alpha$ be curve in $M ⊆ \mathbb R^3$ where $M$ is asurface . If $U$ is unit normal of $M$ restricted to the curve a then $S(\alpha')=-U'$ where $S$ is the shape operator and it's defin8ed by $(S_p(v)=-∇_v(U))$ and $∇_v(U)$ is the covariant derivative of $U$ with respect to $v$ . I tried to use the definition of covariant derivative at a point $p$ which is $U(p+t\alpha')'(0)$ and then apply the chain rule so I get the following $S_p(α')=α'U'(p)$ could any one give a hint ?