Example: One Underlying Variable

Let $g(u_1, u_2) = u_1^2 + u_1 u_2$ and $\mathbf{f}(t) = (\cos t, \sin t)$, so $h(t) = g(\cos t, \sin t) = \cos^2 t + \cos t \sin t$.

By the chain rule: $h'(t) = \frac{\partial g}{\partial u_1}\bigg|_{\mathbf{f}(t)} \cdot (-\sin t) + \frac{\partial g}{\partial u_2}\bigg|_{\mathbf{f}(t)} \cdot (\cos t)$.

We have $\frac{\partial g}{\partial u_1} = 2u_1 + u_2$ and $\frac{\partial g}{\partial u_2} = u_1$. At $\mathbf{f}(t) = (\cos t, \sin t)$: $\frac{\partial g}{\partial u_1} = 2\cos t + \sin t$ and $\frac{\partial g}{\partial u_2} = \cos t$. So $h'(t) = (2\cos t + \sin t)(-\sin t) + \cos t \cdot \cos t = -2\sin t \cos t - \sin^2 t + \cos^2 t$.