Page 89 - Schaum's Outline of Theory and Problems of Advanced Calculus

80 DERIVATIVES [CHAP. 4

Since the logarithm is a continuous function, this can be written
( u= u )
1 u 1
log log e
u a u!0 u ¼ u a
lim 1 þ
by Problem 2.19, Chapter 2, with x ¼ u= u.
d log e du
Then by Problem 4.13, ðlog uÞ¼ a .
a
dx u dx
3
2
4.16. Calculate dy=dx if (a) xy 3x ¼ xy þ 5, (b) e xy þ y ln x ¼ cos 2x.
(a)Differentiate with respect to x,considering y as a function of x.(We sometimes say that y is an implicit
function of x, since we cannot solve explicitly for y in terms of x.) Then
d 3 d 2 d d 2 0 3 0
dx ðxy Þ dx ð3x Þ¼ dx ðxyÞþ dx ð5Þ or ðxÞð3y y Þþðy Þð1Þ 6x ¼ðxÞðy ÞþðyÞð1Þþ 0
2
3
where y ¼ dy=dx. Solving, y ¼ð6x y þ yÞ=ð3xy xÞ.
0
0
d xy d d xy y
0
0
dx dx dx x
ðbÞ ðe Þþ ðy ln xÞ¼ ðcos 2xÞ; e ðxy þ yÞþ þðln xÞy ¼ 2 sin 2x:
2x sin 2x þ xye xy þ y
Solving; y ¼ 2 xy
0
x e þ x ln x
2
2
2
4.17. If y ¼ coshðx 3x þ 1Þ, find (a) dy=dx; ðbÞ d y=dx .
2
(a)Let y ¼ cosh u, where u ¼ x 3x þ 1. Then dy=du ¼ sinh u, du=dx ¼ 2x 3, and
dy dy du 2
dx ¼ du dx ¼ðsinh uÞð2x 3Þ¼ ð2x 3Þ sinhðx 3x þ 1Þ

2
2
d y d dy d du d u du 2
sinh u ¼ sinh u þ cosh u
ðbÞ 2 ¼ ¼ 2
dx dx dx dx dx dx dx
2 2 2 2
¼ðsinh uÞð2Þþðcosh uÞð2x 3Þ ¼ 2 sinhðx 3x þ 1Þþð2x 3Þ coshðx 3x þ 1Þ
2
3
4.18. If x y þ y ¼ 2, find (a) y ; ðbÞ y at the point ð1; 1Þ.
00
0
(a)Differentiating with respect to x, x y þ 2xy þ 3y y ¼ 0 and
2 0
2 0
2xy 1
0
x þ 3xy 2
y ¼ 2 2 ¼ at ð1; 1Þ
2 2
d d 2xy 0 0
00 0 ðx þ 3y Þð2xy þ 2yÞ ð2xyÞð2x þ 6yy Þ
dx dx x þ 3y 2
ðbÞ y ¼ ðy Þ¼ 2 2 ¼ 2 2
ðx þ 3y Þ
3
1
Substituting x ¼ 1, y ¼ 1; and y ¼ ,we find y ¼ .
00
0
2 8
MEAN VALUE THEOREMS
4.19. Prove Rolle’s theorem.
Case 1: f ðxÞ 0in ½a; bŠ. Then f ðxÞ¼ 0for all x in ða; bÞ.
0
Case 2: f ðxÞ 6 0in ½a; bŠ.Since f ðxÞ is continuous there are points at which f ðxÞ attains its maximum and
minimum values, denoted by M and m respectively (see Problem 3.34, Chapter 3).
Since f ðxÞ 6 0, at least one of the values M; m is not zero. Suppose, for example, M 6¼ 0 and that
f ð Þ¼ M (see Fig. 4-9). For this case, f ð þ hÞ @ f ð Þ.