Aind the deriative of $$ f(x)=\sqrt{x}\left(2 x^{3}+x^{2}-8 x-3\right) $$.

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Find the first order derivative with respect to $$ x $$ for $$ \sqrt{x}(2x^{3}+x^{2}-8x-3) $$.
Evaluate the derivative by following steps:
- step0: Evaluate the derivative:
  $$\frac{d}{dx}\left(\sqrt{x}	imes \left(2x^{3}+x^{2}-8x-3ight)ight)$$
- step1: Calculate the product:
  $$\frac{d}{dx}\left(2x^{3}\sqrt{x}+x^{2}\sqrt{x}-8x\sqrt{x}-3\sqrt{x}ight)$$
- step2: Simplify:
  $$\frac{d}{dx}\left(2x^{\frac{7}{2}}+x^{\frac{5}{2}}-8x^{\frac{3}{2}}-3x^{\frac{1}{2}}ight)$$
- step3: Use differentiation rules:
  $$\frac{d}{dx}\left(2x^{\frac{7}{2}}ight)+\frac{d}{dx}\left(x^{\frac{5}{2}}ight)-\frac{d}{dx}\left(8x^{\frac{3}{2}}ight)-\frac{d}{dx}\left(3x^{\frac{1}{2}}ight)$$
- step4: Calculate:
  $$7x^{\frac{5}{2}}+\frac{5}{2}x^{\frac{3}{2}}-12x^{\frac{1}{2}}-\frac{3}{2x^{\frac{1}{2}}}$$
- step5: Rewrite the expression:
  $$7x^{\frac{5}{2}}+\frac{5x^{\frac{3}{2}}}{2}-12x^{\frac{1}{2}}-\frac{3}{2x^{\frac{1}{2}}}$$
- step6: Reduce fractions to a common denominator:
  $$\frac{7x^{\frac{5}{2}}	imes 2x^{\frac{1}{2}}}{2x^{\frac{1}{2}}}+\frac{5x^{\frac{3}{2}}	imes x^{\frac{1}{2}}}{2x^{\frac{1}{2}}}-\frac{12x^{\frac{1}{2}}	imes 2x^{\frac{1}{2}}}{2x^{\frac{1}{2}}}-\frac{3}{2x^{\frac{1}{2}}}$$
- step7: Transform the expression:
  $$\frac{7x^{\frac{5}{2}}	imes 2x^{\frac{1}{2}}+5x^{\frac{3}{2}}	imes x^{\frac{1}{2}}-12x^{\frac{1}{2}}	imes 2x^{\frac{1}{2}}-3}{2x^{\frac{1}{2}}}$$
- step8: Multiply the terms:
  $$\frac{14x^{3}+5x^{\frac{3}{2}}	imes x^{\frac{1}{2}}-12x^{\frac{1}{2}}	imes 2x^{\frac{1}{2}}-3}{2x^{\frac{1}{2}}}$$
- step9: Multiply the terms:
  $$\frac{14x^{3}+5x^{2}-12x^{\frac{1}{2}}	imes 2x^{\frac{1}{2}}-3}{2x^{\frac{1}{2}}}$$
- step10: Multiply the terms:
  $$\frac{14x^{3}+5x^{2}-24x-3}{2x^{\frac{1}{2}}}$$
- step11: Simplify:
  $$\frac{14x^{3}+5x^{2}-24x-3}{2\sqrt{x}}$$
- step12: Calculate:
  $$\frac{14x^{3}\sqrt{x}+5x^{2}\sqrt{x}-24x\sqrt{x}-3\sqrt{x}}{2x}$$
The derivative of $$ f(x) = \sqrt{x}(2x^{3}+x^{2}-8x-3) $$ with respect to $$ x $$ is $$ \frac{14x^{3}\sqrt{x}+5x^{2}\sqrt{x}-24x\sqrt{x}-3\sqrt{x}}{2x} $$.
The derivative of $$ f(x) = \sqrt{x}(2x^{3} + x^{2} - 8x - 3) $$ with respect to $$ x $$ is $$ \frac{14x^{3}\sqrt{x} + 5x^{2}\sqrt{x} - 24x\sqrt{x} - 3\sqrt{x}}{2x} $$.

Aind the deriative of .

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