I was having some trouble with the calculus in the section 4-4 of Do Carmo's The Differential Geometry of Curves and Surfaces. This deals with the definition of the covariant derivative.
Here we let $U$ be an open subset of $S$, a regular surface, and let $w$ be a smooth vector field on $U$. Let $\alpha = (-\epsilon, \epsilon) \rightarrow U$ be a parametrized smooth curve with $\alpha(0)=p$.
If we assume $U$ is a the image of parametrization $x: \mathbb{R}^{2} \rightarrow U$, we can write think of $\alpha(t)$ as $x(u(t), v(t))$. If we restrict our vector field $w$, to $\alpha$, we can think of composition as $$w(t)= a(u(t), v(t))x_{u}+b(u(t), v(t))x_{v}.$$ Here the $a$ and $b$ are the smooth components of the vector field $w$. From here the text states that
$$ \frac{dw}{dt}= a(x_{uu}u'+x_{uv}v') + b(x_{vu}u'+x_{vv}v') + a'x_{u}+b'x_{v}$$
What's confusing me are the terms $a(x_{uu}u'+x_{uv}v')$ and $b(x_{vu}u'+x_{vv}v')$. I think we are using the product rule from calculus, but can someone show me how to get these the terms mentioned?
All of this is on page 242 of the text
Hint: Use Leibniz property on $w=ax_u+bx_v$ and the chain rule as in $$\frac{d x_u}{dt}=\frac{\partial x_u}{\partial u}u’+\frac{\partial x_u}{\partial v}v’.$$