1) $y' = (\dfrac{x^3}{3}-\dfrac{x^2}{2}+x-5)'$
$=\dfrac{3.x^2}{3}-\dfrac{2.x}{2}+1$
$=x^2-x+1$
.
2) $y'=(2x^5+\dfrac{x}{2}+3)'$
$= 10x^4+\dfrac{1}{2}$
.
3) $y'=(\dfrac{2}{x}-\dfrac{4}{x^2}+\dfrac{5}{x^3}-\dfrac{6}{7x^4})'$
$=\dfrac{-2}{x^2}+\dfrac{8x}{x^4}-\dfrac{15x^2}{x^6}+\dfrac{24x^3}{7x^8}$
$=\dfrac{-2}{x^2}+\dfrac{8}{x^3}-\dfrac{15}{x^4}+\dfrac{24}{7x^5}$
.
4) $y'=[5x^2(3x-1) ]'$
$=(5x^2)'(3x-1)+5x^2(3x-1)'$
$=10x(3x-1)+5x^2.3$
$=30x^2-10x+15x^2$
$=45x^2-10x$
.
5) $y'=[(x^3-3x)(x^4+x^2-1)]'$
$=(x^3-3x)'(x^4+x^2-1)+(x^3-3x)(x^4+x^2-1)'$
$=(3x^2-3)(x^4+x^2-1)+(x^3-3x)(4x^3+2x)$
$=3x^6+3x^4-3x^2-3x^4-3x^2+3+4x^6+2x^4-12x^4-6x^2$
$=7x^6-10x^4-12x^2+3$
.
6) $y'=[(x^2+5)^3]'$
$=3(x^2+5)^2.(x^2+5)'$
$=3(x^2+5)^2.2x$
$=6x.(x^4+10x^2+25)$
$=6x^5+60x^3+150x$
.
7) $y'=[(x^2+1)(5-3x^2)]'$
$=(x^2+1)'(5-3x^2)+(x^2+1)(5-3x^2)'$
$=2x.(5-3x^2)+(x^2+1)(-6x)$
$=10x-6x^3-6x^3-6x$
$=4x-12x^3$
.
8) $y'=[x(2x-1)(3x+2)]'$
$=x'(2x-1)(3x+2)+x(2x-1)'(3x+2)+x(2x-1)(3x+2)'$
$=(2x-1)(3x+2)+2x(3x+2)+3x(2x-1)$
$=6x^2+x-2+6x^2+4x+6x^2-3x$
$=18x^2+2x-2$
.
9) $y'=[(x+1)(x+2)^2(x+3)^3]'$
$=(x+1)'(x+2)^2(x+3)^3+(x+1).[(x+2)^2]'(x+3)^3+(x+1)(x+2)^2[(x+3)^3]'$
$=(x+2)^2(x+3)^3+(x+1).2(x+2).(x+2)'(x+3)^3+(x+1)(x+2)^2.3(x+3)^2.(x+3)'$
$=(x+2)^2(x+3)^3+(x+1).2(x+2)(x+3)^3+(x+1)(x+2)^2.3(x+3)^2$
.
