Определение:
![\lim_{dx \to 0} \frac{dy}{dx} = \lim_{dx \to0} \frac{f(x+dx)-f(x)}{dx}](https://tex.z-dn.net/?f=+%5Clim_%7Bdx+%5Cto+0%7D++%5Cfrac%7Bdy%7D%7Bdx%7D+%3D+%5Clim_%7Bdx+%5Cto0%7D++%5Cfrac%7Bf%28x%2Bdx%29-f%28x%29%7D%7Bdx%7D++)
нахождение производной:
у=х²+1
f(x)=x²+1
f(x+dx)=(<span>x+dx)</span>²+1=x²+2xdx+(dx)²+1
![\lim_{dx \to0} \frac{f(x+dx)-f(x)}{dx}=\lim_{dx \to0} \frac{x^2+2xdx+(dx)^2+1-(x^2+1)}{dx}= \\ \\ =\lim_{dx \to0} \frac{x^2+2xdx+(dx)^2+1-x^2-1)}{dx}= \lim_{dx \to0} \frac{2xdx+(dx)^2}{dx}= \\ \\ =\lim_{dx \to0} \frac{dx(2x+dx)}{dx}= \lim_{dx \to0} (2x+dx)=2x+0=2x](https://tex.z-dn.net/?f=%5Clim_%7Bdx+%5Cto0%7D+%5Cfrac%7Bf%28x%2Bdx%29-f%28x%29%7D%7Bdx%7D%3D%5Clim_%7Bdx+%5Cto0%7D+%5Cfrac%7Bx%5E2%2B2xdx%2B%28dx%29%5E2%2B1-%28x%5E2%2B1%29%7D%7Bdx%7D%3D+%5C%5C++%5C%5C+%3D%5Clim_%7Bdx+%5Cto0%7D+%5Cfrac%7Bx%5E2%2B2xdx%2B%28dx%29%5E2%2B1-x%5E2-1%29%7D%7Bdx%7D%3D++%5Clim_%7Bdx+%5Cto0%7D+%5Cfrac%7B2xdx%2B%28dx%29%5E2%7D%7Bdx%7D%3D++%5C%5C++%5C%5C+%3D%5Clim_%7Bdx+%5Cto0%7D+%5Cfrac%7Bdx%282x%2Bdx%29%7D%7Bdx%7D%3D+%5Clim_%7Bdx+%5Cto0%7D+%282x%2Bdx%29%3D2x%2B0%3D2x)
Sin⁴xCos²x - Cos⁴xSin²x = Cos2x
Sin²xCos²x(Sin²x - Cos²x) - Cos2x = 0
Sin²xCos²x * (- Cos2x) - Cos2x = 0
Cos2x(Sin²xCos²x + 1) = 0
Cos2x = 0 Sin²xCos²x + 1 = 0
2x = π/2 + πn , n ∈ z 1/4Sin²2x + 1 = 0
x = π/4 + πn/2 , n ∈ z Sin²2x = - 4 - решений нет
Ответ : π/4 + πn/2 , n ∈ z
7/Задание
№ 4:
Назовите такое значение параметра a, при котором неравенство ax>7x+2 не имеет решений.
РЕШЕНИЕ:
ax>7x+2
ax-7x>2
(a-7)x>2
Если а=7, то неравенство
0>2 не имеет решений.
Если а>7, то решения x>2/(a-7)
Если а<7, то решения x<2/(a-7)
ОТВЕТ: 7
Otvet))))))))))))))))))))))))