1 Derivacije

Napomena. U svakom zadatku su prije racunanja derivacije definirane funkcije zasebno,
kako biste zorno jasnije vidjeli o kojoj se funkciji radi. Naravno da nema potrebe funkciju
zasebno definirati, moze se njezina formula direktno unositi u naredbu "diff".

1. zadatak

(%i1) f(x):=-3/4*x^4+5/3*x^3-3/8*x^2+5/7*x+2/3;
(%o1) f(x):=2/3+(5*x)/7+(-3*x^2)/8+(5*x^3)/3+(-3*x^4)/4
(%i2) diff(f(x),x);
(%o2) -3*x^3+5*x^2-(3*x)/4+5/7

2. zadatak

(%i3) f(x):=-4/(7*x^5);
(%o3) f(x):=(-4)/(7*x^5)
(%i4) diff(f(x),x);
(%o4) 20/(7*x^6)

3. zadatak

(%i5) f(x):=(2*x^5-4*x^4+9*x^3+3*x^2)/(7*x^4);
(%o5) f(x):=(3*x^2+9*x^3-4*x^4+2*x^5)/(7*x^4)
(%i6) diff(f(x),x);
(%o6) (6*x+27*x^2-16*x^3+10*x^4)/(7*x^4)-(4*(3*x^2+9*x^3-4*x^4+2*x^5))/(7*x^5)
(%i7) ratsimp(%);
(%o7) (-6-9*x+2*x^3)/(7*x^3)

4. zadatak

(%i8) f(x):=(3*x-sqrt(x))*(4*x^(1/3)+5*x^(3/4));
(%o8) f(x):=(3*x-sqrt(x))*(5*x^(3/4)+4*x^(1/3))
(%i9) diff(f(x),x);
(%o9) (4/(3*x^(2/3))+15/(4*x^(1/4)))*(3*x-sqrt(x))+(3-1/(2*sqrt(x)))*(4*x^(1/3)+5*x^(3/4))
(%i10) expand(%);
(%o10) (105*x^(3/4))/4+16*x^(1/3)-(25*x^(1/4))/4-10/(3*x^(1/6))

5. zadatak

(%i11) f(x):=x*%e^x;
(%o11) f(x):=x*%e^x
(%i12) izraz:diff(f(x),x);
(izraz) x*%e^x+%e^x
(%i13) subst(0,x,izraz);
(%o13) 1

6. zadatak

(%i14) f(x):=x^(1/3)*log(x)/log(2);
(%o14) f(x):=(x^(1/3)*log(x))/log(2)
(%i15) diff(f(x),x);
(%o15) log(x)/(3*log(2)*x^(2/3))+1/(log(2)*x^(2/3))

7. zadatak

(%i16) f(x):=(x^2-3*x)/(x^2-3*x+2);
(%o16) f(x):=(x^2-3*x)/(2-3*x+x^2)
(%i17) diff(f(x),x);
(%o17) (2*x-3)/(x^2-3*x+2)-((2*x-3)*(x^2-3*x))/(2-3*x+x^2)^2
(%i18) ratsimp(%);
(%o18) (4*x-6)/(x^4-6*x^3+13*x^2-12*x+4)

8. zadatak

(%i19) f(x):=3/(4*x^3-5*x^2+7*x-1);
(%o19) f(x):=3/(-1+7*x-5*x^2+4*x^3)
(%i20) diff(f(x),x);
(%o20) -(3*(7-10*x+12*x^2))/(-1+7*x-5*x^2+4*x^3)^2

9. zadatak

(%i21) f(x):=(2*x^(2/5)-1)/(2*x^(2/5)+1);
(%o21) f(x):=(2*x^(2/5)-1)/(1+2*x^(2/5))
(%i22) diff(f(x),x);
(%o22) 4/(5*(2*x^(2/5)+1)*x^(3/5))-(4*(2*x^(2/5)-1))/(5*(1+2*x^(2/5))^2*x^(3/5))
(%i23) radcan(%);
(%o23) 8/(20*x^(7/5)+20*x+5*x^(3/5))

10. zadatak

(%i24) f(x):=4/x^2-3/x+2*x/3;
(%o24) f(x):=(2*x)/3-3/x+4/x^2
(%i25) der:diff(f(x),x);
(der) 3/x^2-8/x^3+2/3
(%i26) [subst(3,x,der),subst(-3,x,der)];
(%o26) [19/27,35/27]

f'(3)+f'(-3)=2

(%i27) subst(3,x,der)+subst(-3,x,der);
(%o27) 2

11. zadatak

(%i28) f(x):=x^3/(1+%e^x);
(%o28) f(x):=x^3/(%e^x+1)
(%i29) diff(f(x),x);
(%o29) (3*x^2)/(%e^x+1)-(x^3*%e^x)/(1+%e^x)^2
(%i30) ratsimp(%);
(%o30) -((x^3-3*x^2)*%e^x-3*x^2)/(%e^(2*x)+2*%e^x+1)

12. zadatak

(%i31) f(x):=10^x*log(x)/log(10)+log(10);
(%o31) f(x):=log(10)+(10^x*log(x))/log(10)
(%i32) diff(f(x),x);
(%o32) 10^x*log(x)+10^x/(log(10)*x)

13. zadatak

(%i33) f(x):=(log(x)+1)/(log(x)-1);
(%o33) f(x):=(1+log(x))/(log(x)-1)
(%i34) diff(f(x),x);
(%o34) 1/(x*(log(x)-1))-(1+log(x))/(x*(log(x)-1)^2)
(%i35) ratsimp(%);
(%o35) -2/(x*log(x)^2-2*x*log(x)+x)

14. zadatak

(%i36) f(x):=(sin(x)-tan(x))/(cos(x)+tan(x));
(%o36) f(x):=(sin(x)-tan(x))/(tan(x)+cos(x))
(%i37) diff(f(x),x);
(%o37) (cos(x)-sec(x)^2)/(tan(x)+cos(x))-((sec(x)^2-sin(x))*(sin(x)-tan(x)))/(cos(x)+tan(x))^2
(%i38) trigsimp(%);
(%o38) (-2*cos(x)+cos(x)^2+cos(x)^3+(cos(x)^2-1)*sin(x))/(sin(x)^2+2*cos(x)^2*sin(x)+cos(x)^4)

15. zadatak

(%i39) f(x):=(cos(x)-sin(x))*tan(x);
(%o39) f(x):=(cos(x)-sin(x))*tan(x)
(%i40) diff(f(x),x);
(%o40) (-cos(x)-sin(x))*tan(x)+sec(x)^2*(cos(x)-sin(x))
(%i41) trigsimp(%);
(%o41) -((cos(x)^2+1)*sin(x)-cos(x)^3)/cos(x)^2
(%i42) expand(%);
(%o42) -sin(x)/cos(x)^2-sin(x)+cos(x)

2 Derivacije složenih funkcija

1. zadatak

(%i43) f(x):=(x^2+3*x-5)^20;
(%o43) f(x):=(x^2+3*x-5)^20
(%i44) diff(f(x),x);
(%o44) 20*(2*x+3)*(-5+3*x+x^2)^19

2. zadatak

(%i45) f(x):=(3*x-1)^(4/5);
(%o45) f(x):=(3*x-1)^(4/5)
(%i46) diff(f(x),x);
(%o46) 12/(5*(3*x-1)^(1/5))

3. zadatak

(%i47) f(x):=sqrt((x^2-3)/(x^2+3));
(%o47) f(x):=sqrt((x^2-3)/(3+x^2))
(%i48) diff(f(x),x);
(%o48) x/(sqrt(x^2-3)*sqrt(x^2+3))-(x*sqrt(x^2-3))/(3+x^2)^(3/2)
(%i49) ratsimp(%);
(%o49) (6*x*sqrt(x^2+3))/(sqrt(x^2-3)*(x^4+6*x^2+9))

4. zadatak

(%i50) f(x):=(3*x-5*x^4)*%e^(2*x);
(%o50) f(x):=(3*x-5*x^4)*%e^(2*x)
(%i51) diff(f(x),x);
(%o51) 2*(3*x-5*x^4)*%e^(2*x)+(3-20*x^3)*%e^(2*x)
(%i52) factor(%);
(%o52) -(-3-6*x+20*x^3+10*x^4)*%e^(2*x)

5. zadatak

(%i53) f(x):=log(x^2-4*x)/log(10);
(%o53) f(x):=log(x^2-4*x)/log(10)
(%i54) diff(f(x),x);
(%o54) (2*x-4)/(log(10)*(x^2-4*x))

6. zadatak

(%i55) f(x):=4^(log(x)+5*x^2);
(%o55) f(x):=4^(log(x)+5*x^2)
(%i56) diff(f(x),x);
(%o56) log(4)*(10*x+1/x)*4^(5*x^2+log(x))

7. zadatak

(%i57) f(x):=%e^sqrt(x);
(%o57) f(x):=%e^sqrt(x)
(%i58) diff(f(x),x);
(%o58) %e^sqrt(x)/(2*sqrt(x))

8. zadatak

(%i59) f(x):=log(x)^5;
(%o59) f(x):=log(x)^5
(%i60) diff(f(x),x);
(%o60) (5*log(x)^4)/x

9. zadatak

(%i61) f(x):=log(x^5);
(%o61) f(x):=log(x^5)
(%i62) diff(f(x),x);
(%o62) 5/x

10. zadatak

(%i63) f(x):=log(sin(x));
(%o63) f(x):=log(sin(x))
(%i64) diff(f(x),x);
(%o64) cos(x)/sin(x)
(%i65) trigreduce(%);
(%o65) cot(x)

11. zadatak

(%i66) f(x):=log((x-1)/(x+1));
(%o66) f(x):=log((x-1)/(1+x))
(%i67) diff(f(x),x);
(%o67) ((1+x)*(1/(x+1)-(x-1)/(1+x)^2))/(x-1)
(%i68) ratsimp(%);
(%o68) 2/(x^2-1)

12. zadatak

(%i69) f(x):=7^(sin(x)^2);
(%o69) f(x):=7^sin(x)^2
(%i70) diff(f(x),x);
(%o70) 2*log(7)*7^sin(x)^2*cos(x)*sin(x)

13. zadatak

(%i71) f(x):=log(log(x-3*x^2));
(%o71) f(x):=log(log(x-3*x^2))
(%i72) diff(f(x),x);
(%o72) (1-6*x)/((x-3*x^2)*log(x-3*x^2))

14. zadatak

(%i73) f(x):=2^(2^x);
(%o73) f(x):=2^2^x
(%i74) diff(f(x),x);
(%o74) log(2)^2*2^(x+2^x)

3 Derivacije višeg reda

1. zadatak

(%i75) f(x):=x^(2/3);
(%o75) f(x):=x^(2/3)

prva derivacija

(%i76) diff(f(x),x);
(%o76) 2/(3*x^(1/3))

druga derivacija

(%i77) diff(f(x),x,2);
(%o77) -2/(9*x^(4/3))

treća derivacija

(%i78) der3:diff(f(x),x,3);
(der3) 8/(27*x^(7/3))

vrijednost treće derivacije u točki x=8

(%i79) subst(8,x,der3);
(%o79) 1/432

2. zadatak - četvrta derivacija funkcije

(%i80) f(x):=2/(x+1);
(%o80) f(x):=2/(1+x)
(%i81) diff(f(x),x,4);
(%o81) 48/(1+x)^5
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