Differentiate the quotient by simplifying it algebraically first. $$ h(x)=\frac{2 x^{\frac{1}{2}}-5 x^{\frac{3}{2}}+7 x^{\frac{5}{2}}-9 x^{\frac{7}{2}}}{x^{\frac{1}{2}}} $$

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Calculate or simplify the expression $$ \frac{d}{dx}(\frac{2x^{\frac{1}{2}}-5x^{\frac{3}{2}}+7x^{\frac{5}{2}}-9x^{\frac{7}{2}}}{x^{\frac{1}{2}}}) $$.
Evaluate the derivative by following steps:
- step0: Solution:
  $$\frac{d}{dx}\left(\frac{2x^{\frac{1}{2}}-5x^{\frac{3}{2}}+7x^{\frac{5}{2}}-9x^{\frac{7}{2}}}{x^{\frac{1}{2}}}\right)$$
- step1: Divide the terms:
  $$\frac{d}{dx}\left(2-5x+7x^{2}-9x^{3}\right)$$
- step2: Use differentiation rules:
  $$\frac{d}{dx}\left(2\right)-\frac{d}{dx}\left(5x\right)+\frac{d}{dx}\left(7x^{2}\right)-\frac{d}{dx}\left(9x^{3}\right)$$
- step3: Find the derivative:
  $$0-5+14x-27x^{2}$$
- step4: Remove 0:
  $$-5+14x-27x^{2}$$
The derivative of the given quotient function $$h(x)$$ is $$-5+14x-27x^{2}$$.
The derivative of $$ h(x) $$ is $$ -5 + 14x - 27x^{2} $$.

Differentiate the quotient by simplifying it algebraically first.

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