33. Find the first derivative of the following functions.(a)f(x) = ln(xπ+eπ)(b)g(x) =1√πx+x4(c)h(x) =xe1/x2+ln(2x)√x(d)u(x) =xpe-x+√x(e)v(x) = ln(1 +x2)87x2
4. Ift= (x+ 1)(x+ 2)2(x+ 3)3and lnt=ey, finddydxin terms ofxonly.5. Supposef0(2) = 13,g(7) = 2 andg0(7) = 53. Ify=f(g(x)), finddydxx=7.6. Supposey=f(x) is a smooth function. Determine whether each of the followingstatements is true or false.
(a) Iffis increasing, thenf00(x)>0 for allx.(b) Iff0(x)>0 for allx, thenf(x)>0 for allx.(c) Iff0(x)>0 for allx, thenf0is increasing.(d) Iff(x)>0 for allx, thenf0(x)>0 for allx.(e) Iff0(3) = 0, thenfmust have a maximum or a minimum atx= 3.1.3Applications of Derivatives1. IfLis a tangent line to the graphy=x2+ 3, and thex-intercept ofLis 1, find allpossible points at whichLtouches the graph ofy=x2+ 3.2. The positionxof a particle at timetis given byx=t3+at2+bt+cfor some constantsa, b, c. It is known that whent= 0, the particle is at positionx= 0; also, there is a certain timet0such that the velocity and acceleration of theparticle are both zero at timet0, and at timet0the particle is at positionx= 1.Find the values ofa,bandc.