Първата и втората производна

[avii]

f(x)=[tex](x-1)*\sqrt[3]{x^2}[/tex]

[math10.com]

[tex]\sqrt[3]{x^2}=x^{\frac{2}{3}} ; =>f(x)=(x-1)\sqrt[3]{x^2}=x^{\frac{5}{3}}-x^{\frac{2}{3}}[/tex]

[tex]f'(x)=(x^{\frac{5}{3}})'-(x^{\frac{2}{3}})'=\frac{5}{3}.x^{\frac{5}{3}-1}-\frac{2}{3}.x^{\frac{2}{3}-1}=\frac{5}{3}.x^{\frac{2}{3}}-\frac{2}{3}.x^{-\frac{1}{3}}=\frac{5\sqrt[3]{x^2}}{3}-\frac{2}{3\sqrt[3]{x}}=\frac{5x\sqrt[3]{x^2}-2\sqrt[3]{x^2}}{3x}=\frac{(5x-2)\sqrt[3]{x^2}}{3x}[/tex]

[tex]f''(x)=(x^{\frac{5}{3}})'-(x^{\frac{2}{3}})'=\frac{5}{3}.\frac{2}{3}.x^{-\frac{1}{3}}-\frac{2}{3}.\frac{-1}{3}.x^{-\frac{4}{3}}=\frac{10x^{\frac{2}{3}}}{9x}+\frac{2x^{\frac{2}{3}}}{9x^2}=\frac{2(5x+1)x^{\frac{2}{3}}}{9x^2}=\frac{2(5x+1)\sqrt[3]{x^2}}{9x^2}[/tex]