A function F:ℝⁿ→ℝ^m is differentiable at the point p in ℝⁿ if there exists a linear map dF_p:ℝⁿ→ℝ^m such that

dF_p is known as the differential (or derivative) of F at p. When m = n = 1, then dF_p is a 1×1 matrix with entry F’(p). More generally dF_p is represented by the Jacobian matrix at p. A function F:ℝⁿ→ℝ^m can be expanded as

It is sufficient for F to be differentiable for all the partial derivatives ∂f_i/∂x_j to exist and be continuous (see continuous function). More generally, for a differentiable function F:M → N between smooth manifolds M and N, the differential at p in M is a linear map dF_p:T_p(M) → T_F(p)(N) between tangent spaces. For v in T_p(M) take a curve γ‎ in M such that γ‎(0) = p and γ‎’ (0) = v. Then F(γ‎) is a curve in N and dF_p(v) = (F◦γ‎)’ (0).