matma6

442 Rozwiązania i odpowiedzi

6.152. z' = l>V.

, 4x² + l ,

6.154. x>0, z — —-— e^x.

4 xjx

6.153. z = e^2x(30x ²+20x—3). 6.155. /=(l+fc²)e“^,rcsin* .

6.156. y' = 5^x ln 5 +2* ln 2 6.158. y' = 2 • 7^X ln 7.

6.157. y' = 3^xx²(xln 3+3). 6.159. >' = 15 • 10^3x In 10.

6.160. a>0, y' = a^2x x"'^l(2x ln a +n).    6.161. x#0, y' = —.

6.162. >' = 70 • 5^10x ln 5 .

,    5

6.164. x^0, >' = —.

x

,    -3

6.166. x#2, z =-- .

x-2

-1

6.163. x?4 — 3, z’ —-- .

x+3

6.165. s' =

7t²+r

6.168. (>|2|, >' =

-2

Vr²-4

i

6.167. t#± 1, s' = --2 •

1 -r

1

6.169. x^0, x^l, y =——j—..

x ln |x|

,    lab

6.170. a cos² x^b sin² x, >' = -=-j-.2 • z •

a cos x—b sin x

¹    6.172. >'=- tg ±x.

6.171. cos x^0, >' = 6.173. cos x 5^0, >' =

cos x 1

cos X

6.174. >'= 8 ctg ⁵x cos x .

6.175. X540, X54l, >' = --—j=

(1 x) yjx

6.176.    >'=    ² - - Wskazówka. Usunąć niewymierność z mianownika.

Vx²+1

6.177.    >' = ctg x.

6.178. x>0, x?il, >'=--—= .

(l-x) Vx

1 —ln(lnx) 1—>lnx 6.180. >'=    ,, ~-₂-= _n~g •

x (ln x) x (ln x)

6.179.    :•

x(a+x)

ln a

6.181. >' = -

x (ln x)²

6.182. x>0, >' = 5x^!x(lnx +1).

6.183. x>0, >' = — 30x~ ^3x (ln x+!)•

Do rozdziału VI

443

6.184. x^0, y' = x*^{in x}^—— +cos xln x^j . 6.185. y' = 3x^CM    _sj_n x ln ,

6.186. -^>0, y' = (^j ^^ln 7^-1)-    6187. ^x>0’ / = ^*"²(l-lnx).

6.188.    a>0, y'=x'^na-^l\na.

Wskazówka. a=e'^aa, a^,n*=(e^lnx)^,n*=(7°*).

6.189.    y' = 5^,o2ln 5x^)nS-¹ .

6.190.    / = 0. Uwaga. Pochodna y' = 0, ponieważ y=x^l,,a x=(e^{,n x})^I/ln x=e.

(cos² x    \

—r--sin xln sin x ).

sin x    J

6.192. x>0, y'=(arctg x)^x ( ln arctg x +

)

(1 +x²) arctg x)

--l-cos x ln tg xY

cos x    /

i- /    1    sin x    \

'* -j—¹--ź— ln tgx).

\sin x cos"¹ x cos x /

6.193. tgx>0, y'=(tgx)^sinx(-+cosxln

6.194. tgx>0, y' = (tgx)^c

: cos x cos

6.195.    cosx>0, sinx^0, y'= —(cos x)^{clt x}|° ^C°^S * + l^ •

\ sin x )

6.196.    y' = e^x+'*.

6.197.    Jeżeli x>0, to x=e^lnx, y = x*'=(e^lnx)**=e" ^,nx,

y' = e'^{x ,nx} (f In x + e" i «•*    ¹ e^x ^ln x+Y j = e^x+" ^{ln x} ^ln x + ^ .

albo inaczej:

lny = e^xlnx, (lny)’ = — = e^x ^lnx + —y^ye^ln x +Y^ = e"e^x^ln X +—

6.198. Jeżeli x>0, to x=e^lnx, x^x = e^xlnx, y = x^x'^,=e'*^ln*^,n\

albo inaczej:

(e^x ,n xln x)' = X*' (e^xX(x ln x)' ln x + e"^ln *    =

=x^xV^x ^(ln x)² + ln x+i-^=x*^+x-'^(ln x)² + ln x+^ ,

y =e

_ex ln X ln jc/- X Ii

Iny=x^xlnx, (lny)'=—=(x^x)'Inx+x* —,