Упр.8.1 ГДЗ Мерзляк 11 класс Базовый уровень (Алгебра)

1) y=x^v5; 6) y=e^x sin(x); 11) y=7^(2x-3);
2) y=4e^x; 7) y=e^x/(x-2); 12) y=x·3^x;
3) y=e^(5x); 8) y=e^x+e^(-x); 13) y=(2^x-3)/(2^x+1);
4) y=x^3 e^x; 9) y=5^x; 14) y=0,3^tg(x).
5) y=x^v3 e^x; 10) y=2^x^2;

1. $$y=x^{\sqrt5}$$
   $$y’=\sqrt5\,x^{\sqrt5-1}$$
2. $$y=4e^x$$
   $$y’=4e^x$$
3. $$y=e^{5x}$$
   $$y’=5e^{5x}$$
4. $$y=x^3e^x$$
   $$y’=(x^3)’e^x+x^3(e^x)’=3x^2e^x+x^3e^x=x^2e^x(3+x)$$
5. $$y=x^{\sqrt3}e^x$$
   $$y’=(x^{\sqrt3})’e^x+x^{\sqrt3}(e^x)’=\sqrt3\,x^{\sqrt3-1}e^x+x^{\sqrt3}e^x$$
   $$y’=e^x\left(\sqrt3\,x^{\sqrt3-1}+x^{\sqrt3}\right)$$
6. $$y=e^x\sin x$$
   $$y’=(e^x)’\sin x+e^x(\sin x)’=e^x\sin x+e^x\cos x=e^x(\sin x+\cos x)$$
7. $$y=\frac{e^x}{x-2}$$
   $$y’=\frac{(e^x)'(x-2)-e^x(x-2)’}{(x-2)^2}=\frac{e^x(x-2)-e^x}{(x-2)^2}=\frac{e^x(x-3)}{(x-2)^2}$$
8. $$y=e^x+e^{-x}$$
   $$y’=e^x-e^{-x}$$
9. $$y=5^x$$
   $$y’=5^x\ln 5$$
10. $$y=2^{x^2}$$
    $$y’=(x^2)’\cdot 2^{x^2}\ln 2=2x\cdot 2^{x^2}\ln 2$$
11. $$y=7^{2x-3}$$
    $$y’=(2x-3)’\cdot 7^{2x-3}\ln 7=2\cdot 7^{2x-3}\ln 7$$
12. $$y=x\cdot 3^x$$
    $$y’=1\cdot 3^x+x\cdot (3^x)’=3^x+x\cdot 3^x\ln 3$$
    $$y’=3^x(1+x\ln 3)$$
13. $$y=\frac{2^x-3}{2^x+1}$$
    $$y’=\frac{(2^x\ln 2)(2^x+1)-(2^x-3)(2^x\ln 2)}{(2^x+1)^2}$$
    $$y’=\frac{2^x\ln 2\bigl((2^x+1)-(2^x-3)\bigr)}{(2^x+1)^2}=\frac{4\cdot 2^x\ln 2}{(2^x+1)^2}$$
14. $$y=0{,}3^{\tg x}$$
    $$y’ = 0{,}3^{\tg x}\ln 0{,}3\cdot (\tg x)’=\frac{0{,}3^{\tg x}\ln 0{,}3}{\cos^2 x}$$

Ответ

1. $$y’=\sqrt5\,x^{\sqrt5-1}$$
2. $$y’=4e^x$$
3. $$y’=5e^{5x}$$
4. $$y’=x^2e^x(3+x)$$
5. $$y’=e^x\left(\sqrt3\,x^{\sqrt3-1}+x^{\sqrt3}\right)$$
6. $$y’=e^x(\sin x+\cos x)$$
7. $$y’=\frac{e^x(x-3)}{(x-2)^2}$$
8. $$y’=e^x-e^{-x}$$
9. $$y’=5^x\ln 5$$
10. $$y’=2x\cdot 2^{x^2}\ln 2$$
11. $$y’=2\cdot 7^{2x-3}\ln 7$$
12. $$y’=3^x(1+x\ln 3)$$
13. $$y’=\frac{4\cdot 2^x\ln 2}{(2^x+1)^2}$$
14. $$y’=\frac{0{,}3^{\tg x}\ln 0{,}3}{\cos^2 x}$$