17.4. EXERCISES 305

−

(3x2y x3 +2yy2 2xy

)(uv

)= o

(uv

)?

Doing the computations, it follows the left side of the equal sign is of the form(3x2uv+3xu2y+3xu2v+u3y+u3v+ v2

xv2 +2uyv+uv2

)

This is o

(uv

)because it involves terms like uv,u2v, etc. Each term being of degree 2 or

more.

17.4 Exercises1. Use the definition of the derivative to find the 1×1 matrix which is the derivative of

the following functions.

(a) f (t) = t2 + t.

(b) f (t) = t3.

(c) f (t) = t sin(t).

(d) f (t) = ln(t2 +1

).

(e) f (t) = t |t|.

2. Show that if f is a real valued function defined on (a,b) and it achieves a localmaximum at x ∈ (a,b), then D f (x) = 0.

3. Use the above definition of the derivative to prove the product rule for functions of 1variable.

4. Let f (x,y) = xsin(y). Compute the derivative directly from the definition.

5. Let f (x,y) = x2 sin(y). Compute the derivative directly from the definition.

6. Let f (x,y) =

(x2 + y

y2

). Compute the derivative directly from the definition.

7. Let f (x,y) =

(x2y

x+ y2

). Compute the derivative directly from the definition.

8. Let f (x,y) = xα yβ . Show D f (x,y) =(

αxα−1yβ xα βyβ−1)

.

9. Let f (x,y) =

(x2 sin(y)

x2 + y

). Find Df (x,y).

10. Let f (x,y) =√

x 3√

y. Find the approximate change in f when (x,y) goes from (4,8)to (4.01,7.99).