Question

(40 баллов) Найдите производную



$y = {x}^{ - 3} \sqrt[3]{ {x}^{4} \sqrt[6]{x \sqrt[3]{x} } }$

Answer 1

$y=x^{-3}\cdot \sqrt[3]{x^4\sqrt[6]{x\sqrt[3]{x} } }\\\\y=x^{-3}\cdot \sqrt[3]{\sqrt[6]{x^{24}\cdot x \sqrt[3]{x}}}=x^{-3}\cdot \sqrt[3]{\sqrt[6]{x^{25} \sqrt[3]{x}}}=x^{-3}\cdot \sqrt[18]{x^{25} \sqrt[3]{x}}=\\\\=x^{-3}\cdot \sqrt[18]{\sqrt[3]{x^{75}\cdot x}}=x^{-3}\cdot \sqrt[54]{x^{76}}=x^{-3}\cdot x^{\frac{76}{54}}=x^{-3}\cdot x^{\frac{38}{27}}=x^{-\frac{43}{27}}\\\\y'=-\frac{43}{27}\cdot x^{-\frac{70}{27}}=-\frac{43}{27}\cdot \frac{1}{\sqrt[27]{x^{70}}}$

$P.S.\; \; y=x^{-3}\cdot \sqrt[3]{x^4\sqrt[6]{x \sqrt[3]{x}}}=x^{-3}\Big (x^4(x\cdot x^{\frac{1}{3}})^{\frac{1}{6}}\Big )^{\frac{1}{3}}=\\\\=x^{-3}\cdot \Big (x^4\cdot x^{\frac{1}{6}}\cdot x^{\frac{1}{18}}\Big )^{\frac{1}{3}}=x^{-3}\cdot \Big (x^{\frac{76}{18}}\Big )^{ \frac{1}{3}}=x^{-3}\cdot x^{\frac{76}{54}}=x^{-\frac{86}{54}}=x^{-\frac{43}{27}}$