Page 96 - Schaum's Outline of Theory and Problems of Advanced Calculus

CHAP. 4] DERIVATIVES 87

2
4.51. If f ðxÞ¼ x þ 3x, find (a) y; ðbÞ dy; ðcÞ y= x; ðdÞ dy=dx; and (e) ð y dyÞ= x,if x ¼ 1 and
x ¼ :01.
Ans. (a).0501, (b).05, (c)5.01, (d)5, (e).01

4.52. Using differentials, compute approximate values for each of the following: (a) sin 318; ðbÞ lnð1:12Þ,
p ffiffiffiffiffi
(c) 5 36.
Ans. (a)0.515, (b)0.12, (c)2.0125
4.53. If y ¼ sin x,evaluate (a) y; ðbÞ dy. (c)Prove that ð y dyÞ= x ! 0as x ! 0.

DIFFERENTIATION RULES AND ELEMENTARY FUNCTIONS
d d d d d d
4.54. Prove: (a) f f ðxÞþ gðxÞg ¼ f ðxÞþ gðxÞ; ðbÞ f f ðxÞ gðxÞg ¼ f ðxÞ gðxÞ,
dx dx dx dx dx dx

d f ðxÞ gðxÞ f ðxÞ f ðxÞg ðxÞ
0
0
ðcÞ ¼ 2 ; gðxÞ 6¼ 0:
dx gðxÞ
½gðxފ
d 3 2 d 2
4.55. Evaluate (a) fx lnðx 2x þ 5Þg at x ¼ 1; ðbÞ fsin ð3x þ =6Þg at x ¼ 0.
dx dx
Ans. (a)3 ln 4; 3 p ffiffiffi 3
2
ðbÞ
d du d du
u
u
4.56. Derive the formulas: (a) a ¼ a ln a ; a > 0; a 6¼ 1; ðbÞ csc u ¼ csc u cot u ;
dx dx dx dx
d 2 du
tanh u ¼ sech u where u is a differentiable function of x:
ðcÞ
dx dx
d d d d
4.57. Compute (a) tan 1 x; ðbÞ csc 1 x; ðcÞ sinh 1 x; ðdÞ coth 1 x, paying attention to the
dx dx dx dx
use of principal values.
x
4.58. If y ¼ x ,computer dy=dx. [Hint: Take logarithms before differentiating.]
x
Ans. x ð1 þ ln xÞ
1
4.59. If y ¼flnð3x þ 2Þg sin ð2xþ:5Þ , find dy=dx at x ¼ 0:

2ln ln2
Ans: þ p ffiffiffi ðln 2Þ =6
4ln2 3
dy dy du dv
4.60. If y ¼ f ðuÞ, where u ¼ gðvÞ and v ¼ hðxÞ,prove that ¼ assuming f , g; and h are differentiable.
dx du dv dx
2
2
4.61. Calculate (a) dy=dx and (b) d y=dx if xy ln y ¼ 1.
4
3
3
2
Ans. (a) y =ð1 xyÞ; ðbÞð3y 2xy Þ=ð1 xyÞ provided xy 6¼ 1
2
2
4.62. If y ¼ tan x,prove that y ¼ 2ð1 þ y Þð1 þ 3y Þ.
000
3
3
2
2
4.63. If x ¼ sec t and y ¼ tan t,evaluate (a) dy=dx; ðbÞ d y=dx ; ðcÞ d y=dx ,at t ¼ =4.
ffiffiffi ffiffiffi
p p
Ans. (a) 2; ðbÞ 1; ðcÞ 3 2
2
2
d y d x dx 3
4.64. Prove that ¼ , stating precise conditions under which it holds.
dx 2 dy 2 dy
4.65. Establish formulas (a)7, (b) 18, and (c) 27, on Page 71.
MEAN VALUE THEOREMS
4.66. Let f ðxÞ¼ 1 ðx 1Þ 2=3 ,0 @ x @ 2. (a) Construct the graph of f ðxÞ.(b) Explain why Rolle’s theorem is
not applicable to this function, i.e., there is no value for which f ð Þ¼ 0, 0 < < 2.
0