Question

НАЙТИ ПРОИЗВОДНЫЕ , 30 баллов. ПОЖАЛУЙСТА

Answer 1

$1)\; \; y=(2\sqrt{x}-x^3)^5\\\\y'=5\, (2\sqrt{x}-x^3)^4\cdot (2\cdot \frac{1}{2\sqrt{x}}-3x^2)\\\\2)\; \; y=\frac{x^3}{x^2-1}\\\\y'=\frac{3x^2(x^2-1)-x^3\cdot 2x}{(x^2-1)^2}=\frac{x^4-3x^2}{(x^2-1)^2}\\\\3)\; \; y=\frac{2x^2-6}{x-2}\\\\y'=\frac{4x\, (x-2)-(2x^2-6)\cdot 1}{(x-2)^2}=\frac{2x^2-8x+6}{(x-2)^2}$

$4)\; \; y=ln\frac{x}{x-1}\\\\y'=\frac{x-1}{x}\cdot \frac{x-1-x}{(x-1)^2}=\frac{x-1}{x}\cdot \frac{-1}{(x-1)^2}=-\frac{1}{x\, (x-1)}\\\\5)\; \; y=ln(tg3x)\\\\y'=\frac{1}{tg3x}\cdot \frac{1}{cos^23x}\cdot 3=\frac{3}{sin3x\cdot cos3x}=\frac{6}{sin6x}$

Answer 2

1)y'=5(2√x -x³)⁴ * (2√x -x³)' = 5(2√x -x³)⁴ * (1/√x -3x²)

2) y'= (3x²(x² -1) - x³*2x)/(x² -1)² = (3x⁴ -3x² -2x⁴)/(x² -1)² = (x⁴ -3x²)/(x² -1)²

3) (4x(x-2) - (2x² -6)*1) /(x -2)² = (4x² -8x -2x² +6)/(x -2)² =( 2x² -8x +6)/(x-2)²

4) y' =(x-1)/x *(x/(x-1)' = (x-1)/x * (1*(x-1) - x*1)/(x-1)² =- (x-1)/x *1/(x-1)² = -1/x(x-1)

5) y'= 1/tg3x * (tg3x)' = 1/tg3x * 1/Cos²3x * 3 = 3/tg3x*Cos²3x=

=3/(Sin3x/Cos3x * Cos²3x) = 3/Sin3xCos3x = 6/(2Sin3xCos3x)= 6/Sin6x