Найдите производные сложных функций: 13) y = ln tgx; 14) y = ln(x^2 - 2x); 15) y = arctg 2x; 16) y = cos x^3; 17) y = sin(2x-1); 18) y = 6x^5(x^2+1)^3; 19) y = e^(3-2x)*cos(3-2x); 20) y = sin(x/3+3)+2^x; 21) y = sqrt((1-x)/6)-2ln((2-5x)/3); 22) y = 2*sqrt[4](1/(x+2)^3)-5e^((x-4)/5); 23) y = arctg 3x^2 в точке x0 = sqrt(3); 24) y = e^(3x-2) - ln(3x-1) в точке x0 = 2/3

Для нахождения производной сложной функции используется правило: $(f(g(x)))' = f'(g(x)) \cdot g'(x)$. 13) $y = \ln \text{tg}x$ $y' = \frac{1}{\text{tg}x} \cdot (\text{tg}x)' = \frac{1}{\text{tg}x} \cdot \frac{1}{\cos^2 x} = \frac{\cos x}{\sin x} \cdot \frac{1}{\cos^2 x} = \frac{1}{\sin x \cos x} = \frac{2}{\sin 2x}$ 14) $y = \ln(x^2 - 2x)$ $y' = \frac{1}{x^2 - 2x} \cdot (x^2 - 2x)' = \frac{2x - 2}{x^2 - 2x}$ 15) $y = \text{arctg} 2x$ $y' = \frac{1}{1 + (2x)^2} \cdot (2x)' = \frac{2}{1 + 4x^2}$ 16) $y = \cos x^3$ $y' = -\sin x^3 \cdot (x^3)' = -3x^2 \sin x^3$ 17) $y = \sin(2x - 1)$ $y' = \cos(2x - 1) \cdot (2x - 1)' = 2\cos(2x - 1)$ 18) $y = 6x^5(x^2 + 1)^3$ Применяем правило произведения $(uv)' = u'v + uv'$: $y' = (6x^5)'(x^2 + 1)^3 + 6x^5((x^2 + 1)^3)' = 30x^4(x^2 + 1)^3 + 6x^5 \cdot 3(x^2 + 1)^2 \cdot 2x = 30x^4(x^2 + 1)^3 + 36x^6(x^2 + 1)^2 = 6x^4(x^2 + 1)^2(5(x^2 + 1) + 6x^2) = 6x^4(x^2 + 1)^2(11x^2 + 5)$ 19) $y = e^{3-2x} \cdot \cos(3-2x)$ $y' = (e^{3-2x})' \cos(3-2x) + e^{3-2x}(\cos(3-2x))' = -2e^{3-2x} \cos(3-2x) + e^{3-2x}(2\sin(3-2x)) = 2e^{3-2x}(\sin(3-2x) - \cos(3-2x))$ 20) $y = \sin(\frac{x}{3} + 3) + 2^x$ $y' = \cos(\frac{x}{3} + 3) \cdot \frac{1}{3} + 2^x \ln 2 = \frac{1}{3}\cos(\frac{x}{3} + 3) + 2^x \ln 2$ 21) $y = \sqrt{\frac{1-x}{6}} - 2\ln\frac{2-5x}{3}$ $y' = \frac{1}{2\sqrt{\frac{1-x}{6}}} \cdot (-\frac{1}{6}) - 2 \cdot \frac{3}{2-5x} \cdot (-\frac{5}{3}) = -\frac{1}{12\sqrt{\frac{1-x}{6}}} + \frac{10}{2-5x}$ 22) $y = 2 \cdot \sqrt[4]{\frac{1}{(x+2)^3}} - 5e^{\frac{x-4}{5}} = 2(x+2)^{-3/4} - 5e^{\frac{x-4}{5}}$ $y' = 2 \cdot (-\frac{3}{4})(x+2)^{-7/4} - 5e^{\frac{x-4}{5}} \cdot \frac{1}{5} = -\frac{3}{2(x+2)^{7/4}} - e^{\frac{x-4}{5}}$ 23) $y = \text{arctg} 3x^2$ при $x_0 = \sqrt{3}$ $y' = \frac{1}{1 + (3x^2)^2} \cdot 6x = \frac{6x}{1 + 9x^4}$ $y'(\sqrt{3}) = \frac{6\sqrt{3}}{1 + 9(\sqrt{3})^4} = \frac{6\sqrt{3}}{1 + 9 \cdot 9} = \frac{6\sqrt{3}}{82} = \frac{3\sqrt{3}}{41}$ 24) $y = e^{3x-2} - \ln(3x-1)$ при $x_0 = \frac{2}{3}$ $y' = 3e^{3x-2} - \frac{3}{3x-1}$ $y'(\frac{2}{3}) = 3e^{3(\frac{2}{3})-2} - \frac{3}{3(\frac{2}{3})-1} = 3e^0 - \frac{3}{2-1} = 3 \cdot 1 - 3 = 0$