Answer
$11x^{9/2}-\frac{15}{2}x^{3/2}+1$
Work Step by Step
Using the product rule, we obtain
$f'(x)=$$\frac{d}{dx}(x^{3/2})\times(2x^{4}-3x+x^{-1/2})+x^{3/2}\times\frac{d}{dx}(2x^{4}-3x+x^{-1/2})$
$=(\frac{3}{2}x^{1/2})(2x^{4}-3x+x^{-1/2})+x^{3/2}\times(8x^{3}-3-\frac{1}{2}x^{-3/2})$
$=3x^{9/2}-\frac{9}{2}x^{3/2}+\frac{3}{2}+8x^{9/2}-3x^{3/2}-\frac{1}{2}$
$=11x^{9/2}-\frac{15}{2}x^{3/2}+1$
