Упр.199 ГДЗ Колягин Ткачёва 11 класс (Алгебра)

1. $$y=8\sqrt[4]{x}+16e^{x/2}$$

   $$y’ = 8\cdot \frac14 x^{-3/4} + 16\cdot \frac12 e^{x/2}$$
   $$y’ = \frac{2}{\sqrt[4]{x^3}} + 8e^{x/2}$$

2. $$y=\frac{9}{\sqrt[3]{x}}-\frac14\sin 4x$$

   $$y’ = 9\left(-\frac13 x^{-4/3}\right)-\frac14\cdot 4\cos 4x$$
   $$y’=-\frac{3}{x\sqrt[3]{x}}-\cos 4x$$

3. $$y=3x\sqrt[3]{x}-3\ln\frac1x$$

   $$3x\sqrt[3]{x}=3x^{4/3}$$
   $$y’ = 3\cdot \frac43 x^{1/3}-3\left(\ln\frac1x\right)’$$
   $$\left(\ln\frac1x\right)’=\frac{\left(\frac1x\right)’}{\frac1x}=\frac{-1/x^2}{1/x}=-\frac1x$$
   $$y’=4\sqrt[3]{x}+\frac{3}{x}$$

4. $$y=\frac{1}{x\sqrt{x}}+5\cos\frac{x}{5}$$

   $$\frac{1}{x\sqrt{x}}=x^{-3/2}$$
   $$y’=-\frac32 x^{-5/2}+5\left(-\sin\frac{x}{5}\right)\cdot \frac15$$
   $$y’=-\frac{3}{2x^2\sqrt{x}}-\sin\frac{x}{5}$$

Ответ

1) $$y’=\frac{2}{\sqrt[4]{x^3}}+8e^{x/2}$$;
2) $$y’=-\frac{3}{x\sqrt[3]{x}}-\cos 4x$$;
3) $$y’=4\sqrt[3]{x}+\frac{3}{x}$$;
4) $$y’=-\frac{3}{2x^2\sqrt{x}}-\sin\frac{x}{5}$$.