Recall that f is differentiable at p if there exists a linear map T : Rn→ Rm such that khk→0lim kf (p + h

1. Quiz 1: Exercise of Computing the derivative of a function

Definition 1.1. Let U be an open subset of Rⁿand f : U → R^mbe a function. Recall that f is differentiable at p if there exists a linear map T : Rⁿ→ R^m such that

khk→0lim

kf (p + h) − f (p) − T (h)k

khk = 0.

The linear map T is denoted by Df (p) and is called the derivative of f at p.

In this exercise, you are going to experience how to find T.

(1) Let f : R² → R² be the function

f (x, y) = (x²− y², 2xy), (x, y) ∈ R². Take p = (2, 1) and h = (h₁, h₂). Verify that

f (p + h) − f (p) = (4h₁− 2h₂+ h²₁− h²₂, 2h₁+ 4h₂+ 2h₁h₂).

(2) Find a linear map T : R²→ R² and a constant M > 0 such that kf (p + h) − f (p) − T (h)k ≤ M khk²

(3) Show that f is differentiable at p and find the derivative of f at p, i.e. find Df (2, 1).

Now you can try to use Definition 1.1 do some other examples.

(1) Let f : R² → R² be the map

f (x, y) = (ax + by, cx + dy)

where a, b, c, d ∈ R. Let p = (x0, y0). Show that f is differentiable at p and find Df (p).

(2) Let f : R² → R³ be the function

f (x, y) = (x + y³, xy, x²y + y²).

Show that f is differentiable at (1, 1) and find Df (1, 1).

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