Answer
(a) applying the Product Rule: $y'=3x^2+10x+2-\frac{1}{x^2}$
(b) multiplying the factors to produce a sum of simpler terms to
differentiate: $y'=3x^2+10x+2-\frac{1}{x^2}$
Work Step by Step
$y=(x^2+1)(x+5+\frac{1}{x})$
(a) applying the Product Rule:
$y'=f'(x)⋅g(x)+f(x)⋅g'(x)$
$y'=((2)x^{2-1}+0)(x+5+\frac{1}{x})+(x^2+1)((1)x^{1-1}+0+(-1)x^{-1-1})$
$y'=(2x)(x+5+\frac{1}{x})+(x^2+1)(1-\frac{1}{x^2})$
$y'=2x^2+10x+\frac{2x}{x}+x^2-\frac{x^2}{x^2}+1-\frac{1}{x^2}$
$y'=3x^2+10x+2-\frac{1}{x^2}$
(b) multiplying the factors to produce a sum of simpler terms to
differentiate:
$y=(x^2+1)(x+5+\frac{1}{x})$
$y=x^3+5x^2+\frac{x^2}{x}+x+5+\frac{1}{x}$
$y=x^3+5x^2+2x+5+\frac{1}{x}$
Derivating the function using the Power Rule:
$y'=(3)x^{3-1}+(2)(5)x^{2-1}+(1)(2)x^{1-1}+(0)+(-1)x^{-1-1}$
$y'=3x^2+10x+2-x^{-2}$
$y'=3x^2+10x+2-\frac{1}{x^2}$
