Matemàtiques - 2n BAT - Exercicis càlcul de funcions derivades

Calcula les derivades de les funcions següents:


1- `f(x)=sin^2(ln(x^2+5))`


SOLUCIÓ:

    `2·sin(ln(x^2+5))·cos(ln(x^2+5))·1/(x^2+5)·2x`


    `(4x·sin(ln(x^2+5))·cos(ln(x^2+5)))/(x^2+5)`


2- `f(x)=((x+2)·ln(x))/(sqrt(x+1)`


SOLUCIÓ:

    `((1·ln(x)+(x+2)/x)·sqrt(x+1)-(x+2)·ln(x)·1/(2sqrt(x+1)))/(sqrt(x+1)^2`


    `((ln(x)+(x+2)/x)·sqrt(x+1)-((x+2)·ln(x))/(2sqrt(x+1)))/(x+1)`


3- `f(x)=cos(sqrt(x^2+16))`


SOLUCIÓ:

    `-sin(sqrt(x^2+16))·1/(2sqrt(x^2+16))·2x`


    `(-x·sin(sqrt(x^2+16)))/sqrt(x^2+16)`


4- `f(x)=tg^3(sqrt(x^2+2))`


SOLUCIÓ:

    `3·tg^2(sqrt(x^2+2))·1/(cos^2(sqrt(x^2+2)))·1/(2sqrt(x^2+2))·2x`


    `(3x·tg^2(sqrt(x^2+2)))/(sqrt(x^2+2)·cos^2(sqrt(x^2+2)))`


5- `f(x)=(1+cos^2(1-3x))^2`


SOLUCIÓ:

    `2(1+cos^2(1-3x))·2cos(1-3x)·(-sin(1-3x)·(-3))`


    `12(1+cos^2(1-3x))·cos(1-3x)·sin(1-3x))`


6- `f(x)=log_2sqrt(x^2/(x^2-4))`


SOLUCIÓ:

    `1/ln(2)·sqrt((x^2-4)/x^2)·1/(2sqrt(x^2/(x^2-4)))·(2x·(x^2-4)-x^2·2x)/((x^2-4)^2)`


    `1/ln(2)·(x^2-4)/x^2·(x·(x^2-4)-x^2·x)/((x^2-4)^2)`


    `1/ln(2)·1/x^2·(x^3-4x-x^3)/(x^2-4)`


    `1/ln(2)·(-4x)/(x^2·(x^2-4))`


    `1/ln(2)·(-4)/(x·(x^2-4))`


    `(-4)/(ln(2)·(x^3-4x))`


7- `f(x)=sec^2(x^3-2)`


SOLUCIÓ:

    Car recordar que `sec x = 1/cos x`


    `(sec^2(x^3-2))'=(1/cos^2(x^3-2))'=(cos^(-2)(x^3-2))'`


    `-2cos^(-3)(x^3-2)·[-sin(x^3-2)]·3x^2`


    `(6x^2sin(x^3-2))/cos^3(x^3-2)`


8- `f(x)=arc sin(sqrt(3x)/2)`


SOLUCIÓ:

    `1/sqrt(1-(sqrt(3x)/2)^2)·sqrt(3)/2·(sqrt(x))'`


    `sqrt(3)/(2sqrt(1-(3x)/4))·1/(2sqrt(x))`


    `sqrt(3)/(4sqrt(x(1-(3x)/4)))`


    `sqrt(3)/(4sqrt((4x-3x^2)/4))`


    `sqrt(3)/(2sqrt(4x-3x^2))`


9- `f(x)=arc tan((3x+2)/4)`


SOLUCIÓ:

    `1/(1+((3x+2)/4)^2)·3/4`


    `3/(4(1+((3x+2)/4)^2))`


    `3/(4(1+(9x^2+12x+4)/16))`


    `3/(4(16+9x^2+12x+4)/16)`


    `3/((9x^2+12x+20)/4)`


    `12/(9x^2+12x+20)`


10- `f(x)=2^(arc tan x)·sqrt(1-x^2)`


SOLUCIÓ:

    `2^(arc tan x)·ln2·1/(1+x^2)·sqrt(1-x^2)+2^(arc tan x)·1/(2sqrt(1-x^2))·(-2x)`


    `2^(arc tan x)((ln2·sqrt(1-x^2))/(1+x^2)-x/sqrt(1-x^2))`


11- `f(x)=(x^2+3)^(x+5)`


SOLUCIÓ:

    `y=(x^2+3)^(x+5)`


    `lny=ln(x^2+3)^(x+5)`


    `lny=(x+5)·ln(x^2+3)`


    `(y')/y=1·ln(x^2+3)+(x+5)·1/(x^2+3)·2x`


    `y'=y·[ln(x^2+3)+(2x(x+5))/(x^2+3)]`


    `y'=(x^2+3)^(x+5)·[ln(x^2+3)+(2x(x+5))/(x^2+3)]`


12- `f(x)=ln sqrt(lnx)`


SOLUCIÓ:

    `1/sqrt(lnx)·1/(2sqrt(lnx))·1/x`


    `1/(2xlnx)`


13- `f(x)=1/sin^3x`


SOLUCIÓ:

    `(1/sin^(3)x)'=(sin^(-3)x)'`


    `-3sin^(-4)x·cosx`

    `(-3cosx)/sin^4x`



14- `f(x)=(e^(2x)+ln^2x+cos(3x))/2`


SOLUCIÓ:

    `1/2[2e^(2x)+(2lnx)/x-3sin(3x)]`


    `e^(2x)+lnx/x-(3sin(3x))/2`


15- `f(x)=5/(2x+6)^3`


SOLUCIÓ:

    `[5(2x+6)^(-3)]'`


    `5·(-3)(2x+6)^(-4)·2`


    `(-30)/(2x+6)^4`


16- `f(x)=e^(tan(3x))·root(3){x^2+1}`


SOLUCIÓ:

    `e^(tan(3x))·1/cos^2(3x)·3·root(3){x^2+1}+e^(tan(3x))·(root(3){x^2+1})'`


    `e^(tan(3x))(3root(3){x^2+1})/cos^2(3x)+e^(tan(3x))·(root(3){x^2+1})'`


    `e^(tan(3x))(3root(3){x^2+1})/cos^2(3x)+e^(tan(3x))·[(x^2+1)^(1/3)]'`


    `e^(tan(3x))(3root(3){x^2+1})/cos^2(3x)+e^(tan(3x))·1/3(x^2+1)^((-2)/3)·2x`


    `e^(tan(3x))[(3root(3){x^2+1})/cos^2(3x)+(2x)/(3(x^2+1)^(2/3))]`