## Thomas' Calculus 13th Edition

(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: $y'=\frac{11x^6+3x^4+4x^{\frac{9}{4}}+12x^{\frac{1}{4}}}{4x^{\frac{17}{4}}}$
$y=(1+x^2)(x^{\frac{3}{4}}-x^{-3})$ (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 $4x^{\frac{17}{4}}$ $y'=\frac{11(x^\frac{17}{4})x^\frac{7}{4}+3(x^4)+(4x^\frac{9}{4})(1)+(3)(4x^\frac{1}{4})}{4x^\frac{17}{4}}$ $y'=\frac{11x^6+3x^4+4x^\frac{9}{4}+12x^\frac{1}{4}}{4x^\frac{17}{4}}$