Let r(t) = 〈x(t), y(t)〉 trace a plane curve . Assume that x′(t₀) ≠ 0. Show that the slope of the tangent vector r′(t₀) is equal to the slope dy/dx of the curve at r(t₀).

Prove that .

Verify the Sum and Product Rules for derivatives of vector-valued functions.

Verify the Chain Rule for vector-valued functions.

Verify the Product Rule for cross products [Eq. (5)].

Verify the linearity properties

Prove the Substitution Rule (where g(t) is a differentiable scalar function):

Prove that if ∥r(t)∥ ≤ K for t ∈ [a, b], then