Упр.224 ГДЗ Колягин Ткачёва 11 класс (Алгебра)

1) f(х) = х3, х0=1;
2) f(x) = sinx, х0 = пи/4;
3) f(x) = lnx, x0=1;
4) f(x) = ex, x0=ln3;
5) f(x) = 3×2-4x, x0=2;
6) f(x) = корень x — 1/корень x, x0=1.

1. $$f(x)=x^3$$

   $$f'(x)=3x^2$$

   $$f'(1)=3\cdot 1^2=3$$

2. $$f(x)=\sin x$$

   $$f'(x)=\cos x$$

   $$f’\left(\frac{\pi}{4}\right)=\cos\frac{\pi}{4}=\frac{\sqrt{2}}{2}$$

3. $$f(x)=\ln x$$

   $$f'(x)=\frac{1}{x}$$

   $$f'(1)=\frac{1}{1}=1$$

4. $$f(x)=e^x$$

   $$f'(x)=e^x$$

   $$f'(\ln 3)=e^{\ln 3}=3$$

5. $$f(x)=3x^2-4x$$

   $$f'(x)=6x-4$$

   $$f'(2)=6\cdot 2-4=8$$

6. $$f(x)=\sqrt{x}-\frac{1}{\sqrt{x}}$$

   $$f'(x)=\frac{1}{2\sqrt{x}}+\frac{1}{2x^{3/2}}$$

   $$f'(1)=\frac{1}{2\sqrt{1}}+\frac{1}{2\cdot 1^{3/2}}=1$$

Ответ

1) $$3$$; 2) $$\frac{\sqrt{2}}{2}$$; 3) $$1$$; 4) $$3$$; 5) $$8$$; 6) $$1$$.