We start with:
$$\int_C ∇f \cdot dr $$
where \(C\) is a piecewise smooth curve with parameterization \(\textbf{r}(t)\), \(a≤t≤b\), and \(f\) be a function of two or three variables with first-order partial derivatives that exist and are continuous on \(C\). Since \(\textbf{r}'(t) = d\textbf{r}(t)/dt\):
$$\int_C ∇f \cdot dr = \int^a_b ∇f(\textbf{r}(t)) \cdot \textbf{r}'(t) dt $$
According to the chain rule:
$$ \frac{d \ f(x(t),y(t))}{dt} = \frac{\partial f}{\partial x} \frac{dx}{dt} + \frac{\partial f}{\partial y} \frac{dy}{dt} = \langle f_x, f_y \rangle \cdot \langle x'(t), y'(t) \rangle $$
Since \([∇f = \langle f_x, f_y \rangle]\) and \([dr = \langle x'(t), y'(t) \rangle]\):
$$ \frac{d f}{dt} = ∇f \cdot dr $$
This means:
$$\begin{align} \int^a_b ∇f(\textbf{r}(t)) \cdot \textbf{r}'(t) dt &= \int^a_b \frac{d}{dt} f(\textbf{r}(t)) \ dt = [f(\textbf{r}(t))]^{t=a}_{t=b} \\ &= f(\textbf{r}(b)) - f(\textbf{r}(a)) \end{align}$$