Answer
(a): Using the Product Rule - $y'=-5x^{4}+12x^{2}-2x-3$
(b): Multiplying the factors to produce a sum of simpler terms to
differentiate - $y'=-5x^{4}+12x^{2}-2x-3$
Work Step by Step
$y = (3 - x^{2}) (x^{3} - x + 1)$
(a) applying the Product Rule:
$y'=f(x)\cdot\:g'(x) + g(x)\cdot\:f'(x)$
$y'=(3-x^{2})((3)x^{3-1}-(1)x^{1-1}+0) + (x^{3}-x+1)(0-(2)x^{2-1})$
$y'=(3-x^{2})(3x^{2}-1)+(x^{3}-x+1)(-2x)$
$y'=9x^{2}-3-3x^{4}+x^{2}-2x^{4}+2x^{2}-2x$
$y'=-5x^{4}+12x^{2}-2x-3$
(b) multiplying the factors to produce a sum of simpler terms to
differentiate:
$y = (3 - x^{2}) (x^{3} - x + 1)$
$y=3x^{3}+3(-x)+3\cdot \:1-x^2x^3-x^2(-x)-x^2\cdot \:1$
$y=3x^{3}-3x+3-x^{5}+x^{3}-x^{2}$
$y=-x^5+4x^3-x^2-3x+3$
Derivating the function using the Power Rule
$y'=-(5)x^{5-1}+(3)(4)x^{3-1}-(2)x^{2-1}-(1)(3)x^{1-1}+(0)$
$y'=-5x^{4}+12x^{2}-2x-3$
