Упр.38.8 ГДЗ Мерзляк 10 класс Углубленный уровень (Алгебра)

1) f(x)=xvx, x_0=81; 3) f(x)=x^2/x^(1/6), x_0=64.
2) f(x)=v(xvx), x_0=16;

1. $$f(x)=x\sqrt{x}=x^{\frac32}$$
   $$f'(x)=\left(x^{\frac32}\right)’=\frac32x^{\frac12}=\frac32\sqrt{x}$$
   $$f'(81)=\frac32\sqrt{81}=\frac32\cdot 9=\frac{27}{2}=13{,}5$$
2. $$f(x)=\sqrt{x\sqrt{x}}=\sqrt{x\cdot x^{\frac12}}=\sqrt{x^{\frac32}}=x^{\frac34}$$
   $$f'(x)=\left(x^{\frac34}\right)’=\frac34x^{-\frac14}=\frac{3}{4\sqrt[4]{x}}$$
   $$f'(16)=\frac{3}{4\sqrt[4]{16}}=\frac{3}{4\cdot 2}=\frac38$$
3. $$f(x)=\frac{x^2}{\sqrt[6]{x}}=x^{2-\frac16}=x^{\frac{11}{6}}$$
   $$f'(x)=\left(x^{\frac{11}{6}}\right)’=\frac{11}{6}x^{\frac56}$$
   $$f'(64)=\frac{11}{6}\cdot 64^{\frac56}=\frac{11}{6}\cdot 2^5=\frac{11}{6}\cdot 32=\frac{176}{3}$$

Ответ

1) $$13{,}5$$; 2) $$\frac38$$; 3) $$\frac{176}{3}$$.