Can we take gradient of a curve?

Question

Consider the case of planar curves in $\mathbb{R}^2$. They can be described by a function $f(x,y) = 0$. For example, a circle can be described by $x^2+y^2=1$. We can take the gradient of this function $$ \nabla_x f = (\partial_x f, \partial_y f),$$ which would produce a vector field with vectors normal to the curve. Also, we can parameterize the same planar curve using a single parameter, say $\alpha(t)=(\cos(t), \sin(t))$. But writing it this way it becomes unclear how to take the gradient of the curve. Is there a way to write the result in terms of the tangent unit vector or normal unit vector of the curve $\alpha$?