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(1 – x)? du
(27)

(1 – x)?
U = cos - x 2
(1 + x)\'da

(1 + x)}"
sin x (2 + e cos x)
(28)

(1 + e cos x)?
du 3e + (2 + e*) cos x
dac
(1 + e cos x)

du
(29) {sin (a' - x*)}",

x cos (a’ – xo) d x

{sin (a’ – xo)}" du

1 (30) log cos-? (1 – xo),

dar (1 – X“)sin -1 x When a function consists of products and quotients of roots and powers, it is generally most convenient to take the differential of the logarithm, or, as it is usually called, the logarithmic differential of the function. (31)

(a + x)" (b + x)",
log u = m log (a + x) + n log (b + x),
1 du

=

Let u =

n

т

+ a + c

u dx

b + c

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+ 1 .x2

-1 (x - 1)} (x + 1)!!

du
(1 + x)"' di (1 + x)"+1"

(x - 2)
{(x - 1)' (x – 3)"}}}
(x - 2)

(r? - 70 + 1).
(1x – 1 )} (x – 3)
(x + 4)

du

X (x + 4) d x

(x + 2)

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=

x + 2

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(39)

U = - (sin x)" (cos x)", du

(sin x)*-- (cos x)"-1 (m cos* x – n sin æ). du

(40)

U =

1

(sin x)"

(cos ~)*' du (sin

(m cosø « + n sin x). (cos x)*+1

du U = €** sin ra, = eco (a sin rx + r cos rx),

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(41)

dx

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(42)

U = €* (sin ræ)", du

= €4* (sin r ~)*-* (a sin rx + mr cos r«).

Implicit Functions of Two Variables.

If u = 0 be an implicit function of two variables w and

y, then

dy d.

du der du dy

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then

dy dx

(k? y')?
(h? x2)

(51) Let (x* + y) = a'm bøya,

dy

{a* – 2 (x + y)} &

dx {b} + 2 (x + y)} y (52) Let (a + y)' (6R y') – 2* y* = 0,

yo (6% y®)? then

da

dy

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Functions of Two or more Variables.

(و)

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3

du =

(* - y) (53) U =

1.2+ y
du
2 x y
du

2x*y
dx (x2 + yo)? (q? – y?)?? dy (.x2 + y) (22 yo)

2xy (ydx ædy)
(x* + y )* (v.2 - y')?

+ y (54)

+ y du y - X – 2 (wy)! du X - Y - 2 (xy) dw 2x(x + y) dy 2y} (x + y) {y – ® – 2 (wy)"} ydx + {v – Ý – 2(xy))}xdy

2 (wy)} (x + y)

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du =

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1 + (29

y) 1 log

(w? - y)!) 2y

du y (x2 - y2)) dy 2 (ydx xdy)

da

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y (202 y')?

du =

77 - 1

(57) Let u = sin (x"y"),
du y"- cos (r" y") (mydx + nxdy).

ydw xdy
(58) If u = sin - 1

y

y (y? – xo)!

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du =

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{(ay - bx) dx + (cx – ax)dy + (b.x cy) dx}.

(cz ax)

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