Question

QUESTION 9 Apply the techniques of logarithmic differentiation to find the derivative of the given function f(x). [Hint: first take the logarithm of both sides and simplify using properties of logarithms. Then differentiate implicitly.) F) - ** (3x+1) {f(x) = {{x^2 (3x + 1)^3} /(sqrt(x^2 + 4))] } TTTT Paragi Arial 3 (12p *DOQ E T' T. . 15 25 Path: P QUESTION 10 Find the derivative of the following expression involving inverse trigonometric functions. g(x) = arctan x + x arcsin x {g(x) = arctan (x^2) + x arcsin x }

Answer 1

f(n)= x+ (3x+1)3 N22+4 f() = 2*43x+1)3 (x+43% . Let y = f(x)= x2(3x+1)3 (x²+ 4) 1/2

y = 32 (3x+1)3 (x2+4)*2 Taking natural log both sides, eny= ln ( 2² (3x+1) 3 [ (x²+4)12 ]

2 л = ln (*13x+y) - (х+4 12- = = kntx')+(3x+1)* 1 x(x+1) — = 2 n x +1 Ln( 3x+1) — 4 East x+4)

Differentiating both sides with respect to a Ceny)= d [2lnx+3 le(34+) - Žlen 12+4)] seg on the area a les x) + 3 of lal3x+1) I a la Iniz?+4)

20143 d 3 이지 캅뾼는 다 (44) 기 | , ) (b

아 ㅋ 8 10 - + - 30)+ (2) 크 - 4 5 +2 - 2. | H 3+1 +4

: 2134+1)(x+4) +9x(x+4) - H.263x+ I dy =- 213x+1) (x+4) o an

2138°+12x+x++4) + 9x3 +362-523-24 N(3x+1) (x+4) Įdy = y dx

- 6x3+24x+2x4+8+ 683 +362-22 x (3x+1)(x+4) cok V dy = 12x²+x²+60x78 #13x+1)(x+4)

dy = y (12x®+x+60x+8) x13x+1) (x%+4) an dy = x2(3x+1)*. (12x++*++60x+8) dx (3?+ 4392 2(3270)(x?+4)

dy = x(3x+1)+(12x3+x_460x+8) (8%+4)!*Ź dy = 2134+1)%(12x2+x²+60x+8) da (x²+4) /

> dfcu) = X(3x+03C127€ +2460X48) 1. da (2²+4)8/2 = f'(X)= x(34+1)*(1283+x°4602+8) (2²+4) 3/2

10. 90n) - arctan(x) + Karcsin Differentiating both sicles with respect to glema este gra) g'ona de Corc tancve?) + x arcsin (0 graadlasctancx')] + have frase sinen)]

glema fobi z volem cred) + u de care sin(n)) + arcsisla) L (M)

glement (23) + 22x NI-22 arc sin(be) El) al(n- 2x - + Hana H24 - NF-R2 + arcsin (2)