21.5. EXERCISES 453
= limt→0
(45
xy√
5+45
xt +15
√5y2 +
45
ty+4
25t2√
5,25
x√
5+15
y√
5+25
t)
=
(45
xy√
5+15
√5y2,
25
x√
5+15
y√
5).
You see from this example and the above definition that all you have to do is to formthe vector which is obtained by replacing each component of the vector with its directionalderivative. In particular, you can take partial derivatives of vector valued functions and usethe same notation.
Example 21.4.11 Find the partial derivative with respect to x of the function f (x,y,z,w)=(xy2,zsin(xy) ,z3x
)T.
From the above definition, f x (x,y,z) = D1f (x,y,z) =(y2,zycos(xy) ,z3
)T.
21.5 Exercises1. Find the directional derivative of f (x,y,z) = x2y+ z4 in the direction of the vector
(1,3,−1) when (x,y,z) = (1,1,1).
2. Find the directional derivative of f (x,y,z) = sin(x+ y2
)+ z in the direction of the
vector (1,2,−1) when (x,y,z) = (1,1,1).
3. Find the directional derivative of f (x,y,z) = ln(x+ y2
)+ z2 in the direction of the
vector (1,1,−1) when (x,y,z) = (1,1,1).
4. Using the conclusion of Proposition 21.4.6, prove Proposition 21.4.8 from the ge-ometric description of the dot product, the one which says the dot product is theproduct of the lengths of the vectors and the cosine of the included angle which is nolarger than π .
5. Find the largest value of the directional derivative of f (x,y,z) = ln(x+ y2
)+ z2 at
the point (1,1,1).
6. Find the smallest value of the directional derivative off (x,y,z) = xsin
(4xy2
)+ z2 at the point (1,1,1).
7. An ant falls to the top of a stove having temperature T (x,y) = x2 sin(x+ y) at thepoint (2,3). In what direction should the ant go to minimize the temperature? Inwhat direction should he go to maximize the temperature?
8. Find the partial derivative with respect to y of the functionf (x,y,z,w) =
(y2,z2 sin(xy) ,z3x
)T.
9. Find the partial derivative with respect to x of the functionf (x,y,z,w) =
(wx,zxsin(xy) ,z3x
)T.
10. Find ∂ f∂x ,
∂ f∂y , and ∂ f
∂ z for f =