Правила дифференцирования
Производная суммы равна сумме производных:
\(\displaystyle (\color{green}{f(x)}+ \color{blue}{g(x)})^{\prime}=(\color{green}{f(x)})^{\prime}+( \color{blue}{g(x)})^{\prime}{\small .}\)
Производная разности равна разности производных:
\(\displaystyle (\color{green}{f(x)}- \color{blue}{g(x)})^{\prime}=(\color{green}{f(x)})^{\prime}-( \color{blue}{g(x)})^{\prime}{\small .}\)
Константа выносится из-под знака производной:
\(\displaystyle (c\cdot \color{green}{f(x)})^{\prime}=c\cdot (\color{green}{f(x)})^{\prime}, \, c=const{\small .}\)
Производная произведения:
\(\displaystyle (\color{green}{f(x)}\cdot \color{blue}{g(x)})^{\prime}=(\color{green}{f(x)})^{\prime}\cdot \color{blue}{g(x)}+\color{green}{f(x)}\cdot (\color{blue}{g(x)})^{\prime}{\small .}\)
Производная частного:
\(\displaystyle \left(\frac{\color{green}{f(x)}}{\color{blue}{g(x)}}\right)^{\prime}=\frac{(\color{green}{f(x)})^{\prime}\cdot \color{blue}{g(x)}-\color{green}{f(x)}\cdot (\color{blue}{g(x)})^{\prime}}{(\color{blue}{g(x)})^2}{\small .}\)
Производная сложной функции
\(\displaystyle (f(g(x)))^{\prime}=f^{\prime}(g)\cdot g^{\prime}(x){\small .}\)