poch1

56 Difftrtntlaiion of Fund lont fCA. i

564. Find y', if

¥ - Y ** y~j *in* x «*• x.

Solution    In ^x + ln (I — *) —In (I +^*)-{-3 In *Jn * + 2 In co*x;

»•-? ¹ , I-O **    , , » ,_L ^llnr

v^v T* ⁺ i-x"r+?⁺³sn^co,Jt"w whcnce y*-y (s~r=_:7“rT^ ⁺ 3co^lx-^2,anjf)-

565. Find y‘, ii y — (sin jc)*.

Solulioa. Iny —x Inslojr. -i y' — In iln x + xcot r, it‘ — (»ln x)* (In sin x + x cot x).

In the following problcms find y' altcr first taking logs of the funclion y»/(x):

566.	y-(*+l)(2* + l)(3*+l).	574. y- Vx.
567.	y-l.+WCa)"	575. y-x^v~‘.
568.		576. y-x*\
569.	y*-* V jręv	577. y-**'"*.
570.	(*-2 )• ^7 Tf^-iru-ar	578.    y— (cos x)^,ln*. 579.    J--(H-i)'.
571.
572.		580. y—(aretan x)*.

573. y-x*\

Sec. 3. The Derivalives of Functions Nol Represented Explicitly

I*. The tkrlv*llve of an invme luncllon. II a functlon y —/(*) ha* a derivative y_g »6 0, tben llie derivative of the lnvcrsc functlon x-/“'(y) b

I

dx I

*“£

lumpie I. Find the derlvatlve x’_f. il

y-x + lnx.

Solutlon. We luve y_a — | +    hence. /    i_.

2*. The dcrlvallve« ol functlon* (epreacnted parauielrlcally. II a functlon y U relatrd to an argument x by mearu ol a parameter t,

Hien

I *-t(0. \ »-*(0.

*-4.

or, In other notation.

i-

Liamplc 2. Find 3^, II

t

dx'

dt

Solutlon. We flnd

Xr-«lC05f, \

y —o ain I. |

t »tn / and ^maeotl. Whcnce

cot 1.

3*. The derhrathr* ol an Impllcil lunctlon. II the relatlonahlp belween x and y Is given in implłcit loim,

F(x.y)-0.    U)

Ihen to find tłve derlvallvr y],—y' In the slmplełt caaea It It aufOcłent: 1) to calculate the derlvatlve, wlth rcspect to x, ol the lelt aide ol equation (I), taking y as a lunclion ol r, 2) to equ#te IhU derlvatlve to iwo. that U. to put

(2)

and 3) to *ol*e the resulllng equation lor y.

Eiample 3. Find the dcrlvatlvc u\ II

*'+*•—3oxy-0.    O)

Solutlon. Forming the deitvatlve ol the lelt aide ol (3) and cquallug U to rwo, we get

3x* +V2’ —³⁰ ty t