Упр.205 ГДЗ Колягин Ткачёва 11 класс (Алгебра)

1. $$e^{1/x}$$
2. $$e^{-2/x}$$
3. $$\ln(2x-1)$$
4. $$\ln 3x$$
5. $$\tg \frac{x}{2}$$
6. $$\cos 4x$$
7. $$\tg(3x+3)$$
8. $$\sin\left(\frac{2x}{3}+1\right)$$

1. $$y=e^{\frac{1}{x}}$$
   $$y’ = e^{\frac{1}{x}}\left(\frac{1}{x}\right)’ = e^{\frac{1}{x}}\cdot\left(-\frac{1}{x^2}\right) = -\frac{e^{\frac{1}{x}}}{x^2}$$
2. $$y=e^{-\frac{2}{x}}$$
   $$y’ = e^{-\frac{2}{x}}\left(-\frac{2}{x}\right)’ = e^{-\frac{2}{x}}\cdot\frac{2}{x^2} = \frac{2e^{-\frac{2}{x}}}{x^2}$$
3. $$y=\ln(2x-1)$$
   $$y’=\frac{(2x-1)’}{2x-1}=\frac{2}{2x-1}$$
4. $$y=\ln 3x$$
   $$y’=\frac{(3x)’}{3x}=\frac{3}{3x}=\frac{1}{x}$$
5. $$y=\tg\frac{x}{2}$$
   $$y’=\frac{\left(\frac{x}{2}\right)’}{\cos^2\frac{x}{2}}=\frac{1}{2\cos^2\frac{x}{2}}$$
6. $$y=\cos 4x$$
   $$y’=-\sin 4x\cdot 4=-4\sin 4x$$
7. $$y=\tg(3x+3)$$
   $$y’=\frac{(3x+3)’}{\cos^2(3x+3)}=\frac{3}{\cos^2(3x+3)}$$
8. $$y=\sin\left(\frac{2x}{3}+1\right)$$
   $$y’=\cos\left(\frac{2x}{3}+1\right)\cdot\left(\frac{2x}{3}+1\right)’=\frac{2}{3}\cos\left(\frac{2x}{3}+1\right)$$

Ответ

$$1)\; y’=-\frac{e^{\frac{1}{x}}}{x^2};\quad 2)\; y’=\frac{2e^{-\frac{2}{x}}}{x^2};\quad 3)\; y’=\frac{2}{2x-1};\quad 4)\; y’=\frac{1}{x};$$
$$5)\; y’=\frac{1}{2\cos^2\frac{x}{2}};\quad 6)\; y’=-4\sin 4x;\quad 7)\; y’=\frac{3}{\cos^2(3x+3)};\quad 8)\; y’=\frac{2}{3}\cos\left(\frac{2x}{3}+1\right).$$