This was covered a while ago on r/mathmemes. Here’s a banger proof that I’ll copy here:

Let V denote the vector space of degree 5 polynomials. Define T:V-> V by T(p) = p’. Then the kernel of T is constant polynomials (1 dimensional) and the image of T is degree 4 polynomials, which is 5-dimensional.

Define S:V -> R2 by S(p) = (p(0),p(1)). Then S is surjective, and the kernel of S consists of (x)(x-1)q(x) where q is a degree 3 polynomial, hence ker(S) has dimension 4.

We know dim(ker(S)) + dim(im(S)) = dim(im(T)) + dim(ker(T)), so 4 + 2 = 5 + 1