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Степенные функции |
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Правила дифференцирования |
| 1 |
$${\left ( const \right )}' = 0$$ |
1 |
$${(c\cdot u)}'= c\cdot {u}'$$ |
| 2 |
$$({u^{n})}'=n\cdot u^{n-1}\cdot {u}'$$ |
1.a |
$${(\frac{u}{c})}'= \frac{1}{c}\cdot {u}'$$ |
| 2.a |
$${x}'=1$$ |
2 |
$${\left ( u\pm v \right )}'= {u}'\pm{v}'$$ |
| 2.b |
$$({u^{2}})'=2\cdot u\cdot {u}'$$ |
3 |
$${\left ( u\cdot v \right )}'= {u}'\cdot v +u\cdot {v}'$$ |
| 2.c |
$${(\frac{1}{u})}'=-\frac{1}{u^{2}}\cdot {u}'$$ |
4 |
$${(\frac{u}{v})}'= \frac{{u}'\cdot v-u\cdot {v}'}{v^{2}}$$ |
| 2.d |
$${(\sqrt{u})}'= \frac{1}{2\cdot \sqrt{u}}\cdot {u}'$$ |
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Показательные функции |
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Сложная функция |
| 3 |
$${(a^{u})}'=a^{u}\cdot \ln{a}\cdot {u}'$$ |
5 |
$${\left (F \left ( u\left ( x \right ) \right ) \right )}'= {F_{u}}'\cdot {u_{x}}'$$ |
| 3.a |
$${(e^{u})}'= e^{u}\cdot {u}'$$ |
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Логарифмические функции |
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Логарифмическое дифференцирование |
| 4 |
$$({\log_a{u}})'=\frac{1}{u\cdot \ln{a}}\cdot {u}'$$ |
6 |
$$y=f\left ( x \right ) \Rightarrow \ln{y} = \ln{f(x)}$$ |
| 4.a |
$${\left ( \ln{u} \right )}'=\frac{1}{u}\cdot {u}'$$ |
7 |
$$\frac{1}{y}\cdot {y}'={(\ln{f(x)})}'$$ |
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Тригонометрические функции |
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| 5 |
$${\left ( \sin u \right )}'= \cos u\cdot {u}'$$ |
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| 6 |
$${\left ( \cos u \right )}'= -\sin u\cdot {u}'$$ |
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| 7 |
$${(\tan u)}'= \frac{1}{\cos ^{2}u}\cdot {u}'$$ |
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| 8 |
$${(\cot u)}'= \frac{1}{\sin ^{2}u}\cdot {u}'$$ |
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Обратные тригонометрические функции |
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| 9 |
$${(\arcsin u)}'= \frac{1}{\sqrt{1-u^{2}}}\cdot {u}'$$ |
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| 10 |
$${(\arccos u)}'= -\frac{1}{\sqrt{1-u^{2}}}\cdot {u}'$$ |
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| 11 |
$${\left ( \arctan u \right )}'= \frac{1}{1+u^{2}}\cdot {u}'$$ |
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| 12 |
$$({\textrm{arccot} u})'= -\frac{1}{1+u^{2}}\cdot {u}'$$ |
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