Question

Найти производные функций

Answer 1

$1)\; \; y=ln\frac{(x+1)^2}{x-2} +4\sqrt[4]{x^3}\\\\y'=\frac{x-2}{(x+1)^2}\cdot \frac{2(x+1)(x-2)-(x+1)^2}{(x-2)^2}+4\cdot \frac{3}{4}\cdot x^{-1/4}=\\\\=\frac{(x+1)\cdot (2x-4-x-1)}{(x+1)^2(x-2)}+3\cdot \frac{1}{\sqrt[4]{x}}=\frac{x-5}{(x+1)(x-2)}+\frac{3}{\sqrt[4]{x}}$

$2)\; \; y=2^{sin^2\frac{1}{x}}\\\\y'=2^{sin^2\frac{1}{x}}\cdot ln2\cdot (sin^2\frac{1}{x})'=2^{sin^2\frac{1}{x}} \cdot ln2\cdot 2\, sin\frac{1}{x}\cdot cos\frac{1}{x}\cdot \frac{-1}{x^2}=\\\\=-2^{sin^2\frac{1}{x}}\cdot ln2\cdot \frac{1}{x^2}\cdot sin\frac{2}{x}\\\\3)\; \; x^3+y^3-3xy=0\\\\3x^2+3y^2\cdot y'-3y-3x\cdot y'=0\\\\y'\, (3y^2-3x)=3y-3x^2\\\\y'=\frac{y-x^2}{y^2-x}$