Answer
(a) applying the Product Rule: $y'=\frac{11x^6+3x^4+4x^{\frac{9}{4}}+12x^{\frac{1}{4}}}{4x^{\frac{17}{4}}}$
(b) multiplying the factors to produce a sum of simpler terms to
differentiate: $y'=\frac{11x^6+3x^4+4x^{\frac{9}{4}}+12x^{\frac{1}{4}}}{4x^{\frac{17}{4}}}$
Work Step by Step
$y=(1+x^2)(x^{\frac{3}{4}}-x^{-3})$
(a) applying the Product Rule:
$y'=f'(x)⋅g(x)+f(x)⋅g'(x)$
$y'=(0+(2)x^{2-1})(x^{\frac{3}{4}}-x^{-3})+(1+x^2)((\frac{3}{4})x^{\frac{3}{4}-1}-(-3)x^{-3-1})$
$y'=2x(x^\frac{3}{4}-x^{-3})+(1+x^2)(\frac{3}{4}x^{-\frac{1}{4}}+3x^{-4})$
$y'=2x^{\frac{3}{4}+1}-2x^{-3+1}+\frac{3}{4}x^{-\frac{1}{4}}+3x^{-4}+\frac{3}{4}x^{-\frac{1}{4}+2}+3x^{-4+2}$
$y'=2x^{\frac{7}{4}}-2x^{-2}+\frac{3}{4}x^{-\frac{1}{4}}+3x^{-4}+\frac{3}{4}x^{\frac{7}{4}}+3x^{-2}$
$y'=\frac{3x^{\frac{7}{4}}}{4}+\frac{3}{4x^{\frac{1}{4}}}+\frac{1}{x^2}+\frac{3}{x^4}+2x^{\frac{7}{4}}$
The least common multiple among denominators is $4x^{\frac{17}{4}}$
$y'=\frac{(x^{\frac{17}{4}})3x^\frac{7}{4}+(x^{4})3+(4x^\frac{9}{4})1+(4x^{\frac{1}{4}})3+(4x^{\frac{17}{4}})2x^\frac{7}{4}}{4x^\frac{17}{4}}$
$y'=\frac{11x^6+3x^4+4x^\frac{9}{4}+12x^\frac{1}{4}}{4x^\frac{17}{4}}$
(b) multiplying the factors to produce a sum of simpler terms to
differentiate
$y=(1+x^2)(x^{\frac{3}{4}}-x^{-3})$
$y=x^\frac{3}{4}-x^{-3}+x^2x^\frac{3}{4}- x^{-3}x^{2} $
$y=x^\frac{3}{4}-x^{-3}+x^{2+\frac{3}{4}}-x^{-3+2}$
$y=x^\frac{11}{4}+x^\frac{3}{4}-x^{-1}-x^{-3}$
Derivating the function using the Power Rule:
$y'=(\frac{11}{4})x^{\frac{11}{4}-1}+(\frac{3}{4})x^{\frac{3}{4}-1}-(-1)x^{-1-1}-(-3)x^{-3-1}$
$y'=\frac{11x^{\frac{7}{4}}}{4}+\frac{3x^{-\frac{1}{4}}}{4}+x^{-2}+3x^{-4}$
$y'=\frac{11x^{\frac{7}{4}}}{4}+\frac{3}{4x^{\frac{1}{4}}}+\frac{1}{x^2}+\frac{3}{x^4}$
The least common multiple among denominators is $4x^{\frac{17}{4}}$
$y'=\frac{11(x^\frac{17}{4})x^\frac{7}{4}+3(x^4)+(4x^\frac{9}{4})(1)+(3)(4x^\frac{1}{4})}{4x^\frac{17}{4}}$
$y'=\frac{11x^6+3x^4+4x^\frac{9}{4}+12x^\frac{1}{4}}{4x^\frac{17}{4}}$
