Question

Производная
$y = \frac{1}{6} {x}^{5} \sqrt{x \sqrt{ {x}^{4} } }$
$y = {( \frac{1}{2}x - 9) }^{3}$
решить
${( \sin( \alpha ) + \cos( \alpha )) }^{2}$

Answer 1

$1)\; \; y=\frac{1}{6}x^5\sqrt{x\sqrt{x^4}}=\frac{1}{6}x^5\cdot \sqrt{\sqrt{x^2\cdot x^4}}=\frac{1}{6}\, x^5\cdot \sqrt[4]{x^6}=\\\\=\frac{1}{6}\, x^5\cdot x^{\frac{6}{4}}=\frac{1}{6}x^5\cdot x^{\frac{3}{2}}=\frac{1}{6}\cdot x^{\frac{13}{2}}\\\\y'=\frac{1}{6}\cdot \frac{13}{2}\cdot x^{\frac{11}{2}}=\frac{13}{12}\cdot \sqrt{x^{11}}\\\\2)\; \; y=(\frac{1}{2}\, x-9)^3\\\\y'=3\cdot (\frac{1}{2}\, x-9)^2\cdot \frac{1}{2}=\frac{3}{2}\cdot (\frac{1}{2}\, x-9)^2\\\\3)\; \; (sina+cosa)^2=\underbrace {sin^2a+cos^2a}_{1}+\underbrace {2\, sina\, cosa}_{sin2a}=1+sin2a\\\\\\\star \; \; \sqrt[n]{\sqrt[k]{x}}=\sqrt[nk]{x}\; \; \to \; \; \sqrt{\sqrt{x}}=\sqrt[4]{x}\; \; \star$