\( y=3 x^{2}+6 x^{\frac{1}{3}}+\frac{2 x^{3}-7}{3 \sqrt{ } x}, \quad x>0 \) find \( \frac{\mathrm{d} y}{\mathrm{~d} x} \). Give each term in your answer in its simplified form.

Find the first order derivative with respect to \( x \) for \( 3x^2+6x^(1/3)+(2x^3-7)/(3*x^(1/2)) \). Evaluate the derivative by following steps: - step0: Evaluate the derivative: \(\frac{d}{dx}\left(3x^{2}+6x^{\frac{1}{3}}+\frac{\left(2x^{3}-7\right)}{\left(3x^{\frac{1}{2}}\right)}\right)\) - step1: Remove the parentheses: \(\frac{d}{dx}\left(3x^{2}+6x^{\frac{1}{3}}+\frac{2x^{3}-7}{\left(3x^{\frac{1}{2}}\right)}\right)\) - step2: Multiply the terms: \(\frac{d}{dx}\left(3x^{2}+6x^{\frac{1}{3}}+\frac{2x^{3}-7}{3x^{\frac{1}{2}}}\right)\) - step3: Add the terms: \(\frac{d}{dx}\left(\frac{9x^{\frac{5}{2}}+18x^{\frac{5}{6}}+2x^{3}-7}{3x^{\frac{1}{2}}}\right)\) - step4: Use differentiation rules: \(\frac{\frac{d}{dx}\left(9x^{\frac{5}{2}}+18x^{\frac{5}{6}}+2x^{3}-7\right)\times 3x^{\frac{1}{2}}-\left(9x^{\frac{5}{2}}+18x^{\frac{5}{6}}+2x^{3}-7\right)\times \frac{d}{dx}\left(3x^{\frac{1}{2}}\right)}{\left(3x^{\frac{1}{2}}\right)^{2}}\) - step5: Calculate: \(\frac{\left(\frac{45}{2}x^{\frac{3}{2}}+15x^{-\frac{1}{6}}+6x^{2}\right)\times 3x^{\frac{1}{2}}-\left(9x^{\frac{5}{2}}+18x^{\frac{5}{6}}+2x^{3}-7\right)\times \frac{3}{2x^{\frac{1}{2}}}}{\left(3x^{\frac{1}{2}}\right)^{2}}\) - step6: Calculate: \(\frac{\frac{3x^{\frac{1}{3}}\left(45x^{\frac{5}{3}}+30+12x^{\frac{13}{6}}\right)}{2}-\frac{3\left(9x^{\frac{5}{2}}+18x^{\frac{5}{6}}+2x^{3}-7\right)}{2x^{\frac{1}{2}}}}{9x}\) - step7: Calculate: \(\frac{\frac{108x^{\frac{5}{2}}+36x^{\frac{5}{6}}+30x^{3}+21}{2x^{\frac{1}{2}}}}{9x}\) - step8: Multiply by the reciprocal: \(\frac{108x^{\frac{5}{2}}+36x^{\frac{5}{6}}+30x^{3}+21}{2x^{\frac{1}{2}}}\times \frac{1}{9x}\) - step9: Rewrite the expression: \(\frac{3\left(36x^{\frac{5}{2}}+12x^{\frac{5}{6}}+10x^{3}+7\right)}{2x^{\frac{1}{2}}}\times \frac{1}{9x}\) - step10: Reduce the fraction: \(\frac{36x^{\frac{5}{2}}+12x^{\frac{5}{6}}+10x^{3}+7}{2x^{\frac{1}{2}}}\times \frac{1}{3x}\) - step11: Multiply the terms: \(\frac{36x^{\frac{5}{2}}+12x^{\frac{5}{6}}+10x^{3}+7}{2x^{\frac{1}{2}}\times 3x}\) - step12: Multiply the terms: \(\frac{36x^{\frac{5}{2}}+12x^{\frac{5}{6}}+10x^{3}+7}{6x^{\frac{3}{2}}}\) - step13: Simplify: \(\frac{36x^{2}\sqrt{x}+12\sqrt[6]{x^{5}}+10x^{3}+7}{6x^{\frac{3}{2}}}\) - step14: Simplify: \(\frac{36x^{2}\sqrt{x}+12\sqrt[6]{x^{5}}+10x^{3}+7}{6x\sqrt{x}}\) - step15: Calculate: \(\frac{36x^{3}+12x\sqrt[3]{x}+10x^{3}\sqrt{x}+7\sqrt{x}}{6x^{2}}\) The derivative of the given function \( y=3x^{2}+6x^{\frac{1}{3}}+\frac{2x^{3}-7}{3\sqrt{x}} \) with respect to \( x \) is: \[ \frac{36x^{3}+12x\sqrt[3]{x}+10x^{3}\sqrt{x}+7\sqrt{x}}{6x^{2}} \]