Let u 1 , u 2 , u 3 , υ 1 , υ 2 , υ 3 , w 1 , w 2 , and w 3 , be differentiable functions of t . Use Exercise 54 to show that d d t u 1 u 2 u 3 υ 1 υ 2 υ 3 w 1 w 2 w 3 = u ′ 1 u ′ 2 u ′ 3 υ 1 υ 2 υ 3 w 1 w 2 w 3 + u 1 u 2 u 3 υ ′ 1 υ ′ 2 υ ′ 3 w 1 w 2 w 3 + u 1 u 2 u 3 υ 1 υ 2 υ 3 w ′ 1 w ′ 2 w ′ 3
Let u 1 , u 2 , u 3 , υ 1 , υ 2 , υ 3 , w 1 , w 2 , and w 3 , be differentiable functions of t . Use Exercise 54 to show that d d t u 1 u 2 u 3 υ 1 υ 2 υ 3 w 1 w 2 w 3 = u ′ 1 u ′ 2 u ′ 3 υ 1 υ 2 υ 3 w 1 w 2 w 3 + u 1 u 2 u 3 υ ′ 1 υ ′ 2 υ ′ 3 w 1 w 2 w 3 + u 1 u 2 u 3 υ 1 υ 2 υ 3 w ′ 1 w ′ 2 w ′ 3
Let
u
1
,
u
2
,
u
3
,
υ
1
,
υ
2
,
υ
3
,
w
1
,
w
2
,
and
w
3
,
be differentiable functions of t. Use Exercise 54 to show that
d
d
t
u
1
u
2
u
3
υ
1
υ
2
υ
3
w
1
w
2
w
3
=
u
′
1
u
′
2
u
′
3
υ
1
υ
2
υ
3
w
1
w
2
w
3
+
u
1
u
2
u
3
υ
′
1
υ
′
2
υ
′
3
w
1
w
2
w
3
+
u
1
u
2
u
3
υ
1
υ
2
υ
3
w
′
1
w
′
2
w
′
3
Consider the function f(p, q, r) =
w = 2.
p+q
r
with p = v²w³+u, q = w³ + v², and r = uv³+ 1. Evaluate when u = 1, v = 3, and
af
δυ
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Use a formula to express w as a function of t if
w =
4s + 9
and s = et − 1.
w =
Another derivative combination Let F = (f. g, h) and let u be
a differentiable scalar-valued function.
a. Take the dot product of F and the del operator; then apply the
result to u to show that
(F•V )u = (3
a
+ h
az
(F-V)u
+ g
+ g
du
+ h
b. Evaluate (F - V)(ry²z³) at (1, 1, 1), where F = (1, 1, 1).
Precalculus: Mathematics for Calculus (Standalone Book)
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