Упр.303 ГДЗ Колягин Ткачёва 11 класс (Алгебра)

Найти вторую производную функции:

1. $$f(x)=\sin 2x$$;
2. $$f(x)=x^3\sin x$$;
3. $$f(x)=x^4+3x^2-x+1$$;
4. $$f(x)=x^4-3x^3+5x+6$$;
5. $$f(x)=e^{\sin x}$$;
6. $$f(x)=\ln(x^2+1)$$.

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

   $$f'(x)=2\sin x\cos x=\sin 2x$$

   $$f»(x)=2\cos 2x$$

2. $$f(x)=x^3\sin x$$

   $$f'(x)=3x^2\sin x+x^3\cos x$$

   $$f»(x)=6x\sin x+3x^2\cos x+3x^2\cos x-x^3\sin x$$

   $$f»(x)=(6x-x^3)\sin x+6x^2\cos x$$

3. $$f(x)=x^4+3x^2-x+1$$

   $$f'(x)=4x^3+6x-1$$

   $$f»(x)=12x^2+6$$

4. $$f(x)=x^4-3x^3+5x+6$$

   $$f'(x)=4x^3-9x^2+5$$

   $$f»(x)=12x^2-18x$$

5. $$f(x)=e^{\sin x}$$

   $$f'(x)=e^{\sin x}\cos x$$

   $$f»(x)=e^{\sin x}\cos^2 x-e^{\sin x}\sin x$$

   $$f»(x)=e^{\sin x}\left(\cos^2 x-\sin x\right)$$

6. $$f(x)=\ln(x^2+1)$$

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

   $$f»(x)=\frac{2(x^2+1)-4x^2}{(x^2+1)^2}$$

   $$f»(x)=\frac{2(1-x^2)}{(x^2+1)^2}$$

Ответ

$$1)\ f»(x)=2\cos 2x;\quad 2)\ f»(x)=(6x-x^3)\sin x+6x^2\cos x;\quad 3)\ f»(x)=12x^2+6;$$

$$4)\ f»(x)=12x^2-18x;\quad 5)\ f»(x)=e^{\sin x}\left(\cos^2 x-\sin x\right);\quad 6)\ f»(x)=\frac{2(1-x^2)}{(x^2+1)^2}.$$