Thm: f is differentiable at a implies f is continuous at a.

Thm: Let f : R^{n} → R^{m}, f = (f_{1}, ..., f_{m}). f is differentiable at a iff f_{i} : R^{n} → R is differentiable at a for all i = 1, ..., m Thm: If f is differentiable at a then \frac{∂f_{i}}{∂x_{j}} exists for all i, j and Df(a) = the Jacobian evaluated at a. Thm: Let f : R^{n} → R^{m}, f = (f_{1}, ..., f_{m}). If \frac{∂f_{i}}{∂x_{j}} exists and are continuous in a neighborhood of a for all i, j, then f is differentiable at a Ex: Is f(x, y) = x^{2}y differentiable at (3, 1).

Find the equation of the tangent plane to f(x, y) = x^{2}y at (3, 1). Estimate f(3.1, .9)

2.4 Thm: If f, g : R^{n} → R^{m} is differentiable at a, then f + g is differentiable at a and D(f + g) = Df + Dg. Thm: Let c ∈ R. If f : R^{n} → R^{m} is differentiable at a, then cf is differentiable at a and D(cf) = cDf. Thm: If g : R^{n} → R^{m} is differentiable at a and if f : R^{m} → R^{k} is differentiable at g(a), then f ∘ g is differentiable at a and D(f ∘ g)(a) = Df(g(a))Dg(a). Note for the product and quotient rule, f, g are real-valued functions, NOT vector valued. Thm: If f, g : R^{n} → R is differentiable at a, then fg is differentiable at a and D(fg) = g(a)Df(a) + f(a)Dg(a). Thm: If g(a) ≠ 0 and f, g : R^{n} → R is differentiable at a, then f/g is differentiable at a and D(f/g) = \frac{g(a)Df(a)−f(a)Dg(a)}{g(a)^{2}}.