Задание 2. Найти производные функций. 1. a) y = xtgx+lncosx+e5x;. 6) y = ex-arcsinx 2. a) y=lnx2x+1+3x3√x; 6) y = 2 arctgr-x2 3. a) y = x2 + x arcsinx + √1-x2;: 6) y=25+1x. 4. a) y = ln(x-1)2x+2+33√x2;. 6) y = 24sinx. 5. a) y = lnx2x-1+4x4√x; 6) y = (esinx +3x)3. 6. a) y=x3 (3lnx-1)-x+1e;: 6) y = (5tg2x+3)4. 7. a) y = ln(x+1)2x+3+3x3√x; 6) y = 5arcsinx2. 8. a) y=e5x(5x-1)-2lnx+17; 6) y = 4 arctg3x. 9. a) y = ln(x+1)2x-2-√4-x2; 6) y = 2 sin1x. 10. a) y = lnx-1+(e3x-7x°)(3x-1); 6) y = 3cos24x

Решим данные примеры на нахождение производных функций. 1. б) \( y = e^{x-\arcsin x} \) \[ y' = e^{x-\arcsin x} \cdot (x-\arcsin x)' = e^{x-\arcsin x} \cdot (1 - \frac{1}{\sqrt{1-x^2}} ) \] 2. б) \( y = 2^{\operatorname{arctg} x - x^2} \) \[ y' = 2^{\operatorname{arctg} x - x^2} \cdot \ln 2 \cdot (\operatorname{arctg} x - x^2)' = 2^{\operatorname{arctg} x - x^2} \cdot \ln 2 \cdot (\frac{1}{1+x^2} - 2x) \] 3. б) \( y = 2^{5 + \frac{1}{x}} \) \[ y' = 2^{5 + \frac{1}{x}} \cdot \ln 2 \cdot (5 + \frac{1}{x})' = 2^{5 + \frac{1}{x}} \cdot \ln 2 \cdot (-\frac{1}{x^2}) \] 4. б) \( y = 2^{\frac{4}{\sin x}} \) \[ y' = 2^{\frac{4}{\sin x}} \cdot \ln 2 \cdot (\frac{4}{\sin x})' = 2^{\frac{4}{\sin x}} \cdot \ln 2 \cdot 4 \cdot ((\sin x)^{-1})' = 2^{\frac{4}{\sin x}} \cdot \ln 2 \cdot 4 \cdot (-1) \cdot (\sin x)^{-2} \cdot \cos x = -4 \cdot 2^{\frac{4}{\sin x}} \cdot \ln 2 \cdot \frac{\cos x}{\sin^2 x} \] 5. б) \( y = (e^{\sin x} + 3x)^3 \) \[ y' = 3 \cdot (e^{\sin x} + 3x)^2 \cdot (e^{\sin x} + 3x)' = 3 \cdot (e^{\sin x} + 3x)^2 \cdot (e^{\sin x} \cdot \cos x + 3) \] 6. б) \( y = (5^{\operatorname{tg} 2x} + 3)^4 \) \[ y' = 4 \cdot (5^{\operatorname{tg} 2x} + 3)^3 \cdot (5^{\operatorname{tg} 2x} + 3)' = 4 \cdot (5^{\operatorname{tg} 2x} + 3)^3 \cdot 5^{\operatorname{tg} 2x} \cdot \ln 5 \cdot (\operatorname{tg} 2x)' = 4 \cdot (5^{\operatorname{tg} 2x} + 3)^3 \cdot 5^{\operatorname{tg} 2x} \cdot \ln 5 \cdot \frac{1}{\cos^2 2x} \cdot 2 = 8 \cdot (5^{\operatorname{tg} 2x} + 3)^3 \cdot 5^{\operatorname{tg} 2x} \cdot \ln 5 \cdot \frac{1}{\cos^2 2x} \] 7. б) \( y = 5^{\arcsin x^2} \) \[ y' = 5^{\arcsin x^2} \cdot \ln 5 \cdot (\arcsin x^2)' = 5^{\arcsin x^2} \cdot \ln 5 \cdot \frac{1}{\sqrt{1 - (x^2)^2}} \cdot 2x = \frac{2x \cdot 5^{\arcsin x^2} \cdot \ln 5}{\sqrt{1 - x^4}} \] 8. б) \( y = 4^{\operatorname{arctg} \frac{3}{x}} \) \[ y' = 4^{\operatorname{arctg} \frac{3}{x}} \cdot \ln 4 \cdot (\operatorname{arctg} \frac{3}{x})' = 4^{\operatorname{arctg} \frac{3}{x}} \cdot \ln 4 \cdot \frac{1}{1 + (\frac{3}{x})^2} \cdot (\frac{3}{x})' = 4^{\operatorname{arctg} \frac{3}{x}} \cdot \ln 4 \cdot \frac{1}{1 + \frac{9}{x^2}} \cdot (-\frac{3}{x^2}) = 4^{\operatorname{arctg} \frac{3}{x}} \cdot \ln 4 \cdot \frac{x^2}{x^2 + 9} \cdot (-\frac{3}{x^2}) = -3 \cdot 4^{\operatorname{arctg} \frac{3}{x}} \cdot \ln 4 \cdot \frac{1}{x^2 + 9} \] 9. б) \( y = 2^{\sin^{\frac{1}{x}} x} \) \[ y' = 2^{\sin^{\frac{1}{x}} x} \cdot \ln 2 \cdot (\sin^{\frac{1}{x}} x)'\] \[ (\sin^{\frac{1}{x}} x)' = (e^{\frac{1}{x} \ln (\sin x)})' = e^{\frac{1}{x} \ln (\sin x)} \cdot (\frac{1}{x} \ln (\sin x))' = \sin^{\frac{1}{x}} x \cdot (\frac{1}{x} \ln (\sin x))' \] \[ (\frac{1}{x} \ln (\sin x))' = (\frac{1}{x})' \cdot \ln (\sin x) + \frac{1}{x} \cdot (\ln (\sin x))' = -\frac{1}{x^2} \ln (\sin x) + \frac{1}{x} \cdot \frac{1}{\sin x} \cdot \cos x = -\frac{\ln (\sin x)}{x^2} + \frac{\cos x}{x \sin x} \] \[ y' = 2^{\sin^{\frac{1}{x}} x} \cdot \ln 2 \cdot \sin^{\frac{1}{x}} x \cdot (\frac{-\ln (\sin x)}{x^2} + \frac{\cos x}{x \sin x}) \] 10. б) \( y = 3^{\cos^2 4x} \) \[ y' = 3^{\cos^2 4x} \cdot \ln 3 \cdot (\cos^2 4x)' = 3^{\cos^2 4x} \cdot \ln 3 \cdot 2 \cos 4x \cdot (\cos 4x)' = 3^{\cos^2 4x} \cdot \ln 3 \cdot 2 \cos 4x \cdot (-\sin 4x) \cdot 4 = -8 \cdot 3^{\cos^2 4x} \cdot \ln 3 \cdot \cos 4x \cdot \sin 4x \]