Рабочая тетрадь по теме Производная

Рабочая тетрадь по теме
«Производная функции и её вычисление»

Рахимова Жанар Муратовна, учитель математики
Майкаинской СОШ №2
Баянаульского района
Павлодарской области

















Майкаин, 2015 г

Тема производная - одна из основных базовых тем, изучаемых в 10 классе. Как и любая другая тема она требует осмысления и хорошего закрепления. Данная рабочая тетрадь предназначена для отработки навыков вычисления первообразной.
По данной теме формул много. Понятно, что запомнить большое количество формул не просто, тем более, что надо не только знать их, но и уметь выбирать самую полезную формулу в конкретной ситуации. Конечно для этого самое реальное средство – практика, решение достаточно большего количества заданий. Данное пособие дает возможность отработать каждую формулу по отдельности. В данном пособии также даны решения на 10 заданий повышенной сложности взятые из сборников тестов 2009-2011 годов. Для проверки уровня усвоенного материала в конце сборника даны тесты на соответствия.
Пособие можно использовать в качестве дополнительного материала на уроках, в качестве домашней работы, а также для самостоятельной работы дома.



Содержание рабочей тетради:
Теоретический материал по данной теме (правила, формулы)
Задания на отработку для каждой формулы уровня А
Задания на отработку уровня В
Разбор сложных заданий уровня С из тестников по подготовке к ЕНТ 2009-2011 годов
Тесты на соответствие, для более прочного усвоения материала
Ответы для самопроверки.


















Производная функции и её вычисление

Операция нахождения производной функции называется дифференцированием.
Определение: Производной функции f в точке х0 называется предел отношения приращения функции f = f(х0+х)- f(х0) к приращению аргумента х при стремлении х к нулю.
Для того, чтобы функция f(х) была дифференцируема в точке х0, необходимо, чтобы она была непрерывна в этой точке.
Правила дифференцирования
Пусть С – постоянная; u, v – функции. Тогда:
(С* u )1 = С* u1
(u + v)1 = u1 + v1
(u * v)1 = u1 * v + v1 *u
(u / v)1 = (u1 * v - v1 *u) / v2

Частные случаи : (u /С)= 1/С * u1
(С/ v) = - С/ v2 * v1

Для нахождения производных используется следующая таблица:

Функция
Производная функции

Функция
Производная

f (x) = С
f1 (х) = 0

f (х) = 13 EMBED Equation.3 1415
f1 (х) = 13 EMBED Equation.3 1415

f (x) = 13 EMBED Equation.3 1415х
f1 (х) = 13 EMBED Equation.3 1415

f (х) = 13 EMBED Equation.3 1415
f1 (х) =13 EMBED Equation.3 1415

f (х) = xn

f1 (х) = 13 EMBED Equation.3 1415

f (х) = 13 EMBED Equation.3 1415
f1 (х) = 13 EMBED Equation.3 1415

f (х) =13 EMBED Equation.3 1415
f1 (х) =13 EMBED Equation.3 1415

f (х) = sin x
f1 (х) = cos x

f (х) =13 EMBED Equation.3 1415
f1 (х) = - 13 EMBED Equation.3 1415

f (х) = cos x

f1 (х) = -sin x

f (х) =13 EMBED Equation.3 1415
f1 (х) =13 EMBED Equation.3 1415

f (х) = tg x
f1 (х) =13 EMBED Equation.3 1415

f (х) =13 EMBED Equation.3 1415
f1 (х) =13 EMBED Equation.3 1415

f (х) = сtg x
f1 (х) =-13 EMBED Equation.3 1415

f (х) = 13 EMBED Equation.3 1415

f1 (х) =13 EMBED Equation.3 1415

f (х) = 13 EMBED Equation.3 1415

f1 (х) =13 EMBED Equation.3 1415

f (х) = 13 EMBED Equation.3 1415

f1 (х) =13 EMBED Equation.3 1415

f (х) = 13 EMBED Equation.3 1415

f1 (х) =-13 EMBED Equation.3 1415

f (х) = 13 EMBED Equation.3 1415
f1 (х) = 13 EMBED Equation.3 1415

f (х) = 13 EMBED Equation.3 1415

f1 (х)= 13 EMBED Equation.3 1415

f (х) = 13 EMBED Equation.3 1415
f1 (х) = 13 EMBED Equation.3 141513 EMBED Equation.3 1415

f (х) = 13 EMBED Equation.3 1415

f1 (х)= -13 EMBED Equation.3 1415



Задания на отработку для каждой формулы уровня А
Используя формулы производных, заполните таблицы:

f (х) = kx f1 (х) = k
f (х)
1/8x
20x
125x
1/3x
1/6x
4,5x
9x
14x

f1 (х)
1/8








f (х) = xn f1 (х) = 13 EMBED Equation.3 1415
f (х)
X5
X8
2X20
813 EMBED Equation.3 1415
5x4
13 EMBED Equation.3 1415
3x14
14x25

f1 (х)
5х4








f (х) = 13 EMBED Equation.3 1415 f1 (х) = 13 EMBED Equation.3 1415
f (х)
3/х
13 EMBED Equation.3 1415
13 EMBED Equation.3 1415
13 EMBED Equation.3 1415
13 EMBED Equation.3 1415
13 EMBED Equation.3 1415
13 EMBED Equation.3 1415
13 EMBED Equation.3 1415

f1 (х)
13 EMBED Equation.3 1415








f (х) = 13 EMBED Equation.3 1415 f1 (х) = - 13 EMBED Equation.3 1415
f (х)
13 EMBED Equation.3 1415
13 EMBED Equation.3 1415
13 EMBED Equation.3 1415
13 EMBED Equation.3 1415
13 EMBED Equation.3 1415
13 EMBED Equation.3 1415

f1 (х)
- 13 EMBED Equation.3 1415






f (х) = 13 EMBED Equation.3 1415 f1 (х) = 13 EMBED Equation.3 1415
f (х)
13 EMBED Equation.3 1415
13 EMBED Equation.3 1415
13 EMBED Equation.3 1415
13 EMBED Equation.3 1415
13 EMBED Equation.3 1415
13 EMBED Equation.3 1415
13 EMBED Equation.3 1415
13 EMBED Equation.3 1415
13 EMBED Equation.3 1415

f1 (х)
-13 EMBED Equation.3 1415









f (х) = 13 EMBED Equation.3 1415 f1 (х) = 13 EMBED Equation.3 1415
f (х)
13 EMBED Equation.3 1415
13 EMBED Equation.3 1415
13 EMBED Equation.3 1415
13 EMBED Equation.3 1415
13 EMBED Equation.3 1415
13 EMBED Equation.3 1415

f1 (х)
13 EMBED Equation.3 1415






f (х) = 13 EMBED Equation.3 1415 f1 (х) = 13 EMBED Equation.3 1415
f (х)
13 EMBED Equation.3 1415
13 EMBED Equation.3 1415
13 EMBED Equation.3 1415
13 EMBED Equation.3 1415
13 EMBED Equation.3 1415
13 EMBED Equation.3 1415

f1 (х)
13 EMBED Equation.3 1415


13 EMBED Equation.3 141513 EMBED Equation.3 1415






f (х) = 13 EMBED Equation.3 1415 f1 (х) =13 EMBED Equation.3 1415
f (х)
13 EMBED Equation.3 1415
13 EMBED Equation.3 1415
13 EMBED Equation.3 1415
13 EMBED Equation.3 1415
13 EMBED Equation.3 1415
13 EMBED Equation.3 1415

f1 (х)
13 EMBED Equation.3 1415






f (х) = 13 EMBED Equation.3 1415 f1 (х) = 13 EMBED Equation.3 1415
f (х)
13 EMBED Equation.3 1415






f1 (х)
13 EMBED Equation.3 1415






f (х) = 13 EMBED Equation.3 1415 f1 (х) = 13 EMBED Equation.3 141513 EMBED Equation.3 1415
f (х)
13 EMBED Equation.3 1415






f1 (х)
13 EMBED Equation.3 141513 EMBED Equation.3 1415






f (х) = 13 EMBED Equation.3 1415 f1 (х) = 13 EMBED Equation.3 1415
f (х)
13 EMBED Equation.3 1415






f1 (х)
13 EMBED Equation.3 1415







f (х) = 13 EMBED Equation.3 1415 f1 (х) = 13 EMBED Equation.3 1415
f (х)
13 EMBED Equation.3 1415
13 EMBED Equation.3 1415
13 EMBED Equation.3 1415
13 EMBED Equation.3 1415
13 EMBED Equation.3 1415
13 EMBED Equation.3 1415

f1 (х)
13 EMBED Equation.3 1415






f (х) = 13 EMBED Equation.3 1415 f1 (х) =13 EMBED Equation.3 1415
f (х)
13 EMBED Equation.3 1415
13 EMBED Equation.3 1415
13 EMBED Equation.3 1415
13 EMBED Equation.3 1415
13 EMBED Equation.3 1415
13 EMBED Equation.3 1415

f1 (х)
13 EMBED Equation.3 1415

13 EMBED Equation.3 1415




f (х) = sin x f1 (х) = cos x
f (х)
3sin x
0.5sin x
sin13 EMBED Equation.3 1415
8sin 13 EMBED Equation.3 1415
5sin3 x
125sin 13 EMBED Equation.3 1415

f1 (х)
3 cos x

13 EMBED Equation.3 1415cos 13 EMBED Equation.3 1415




f (х) = cos x f1 (х) = - sin x
f (х)
cos 3x
2cos 5x
13 EMBED Equation.3 1415cos 13 EMBED Equation.3 1415
cos 20x
cos13 EMBED Equation.3 1415
Cos50 x

f1 (х)
-3sin3 x

-13 EMBED Equation.3 1415sin13 EMBED Equation.3 1415




f (х) = tg x f1 (х) = 13 EMBED Equation.3 1415
f (х)
tg 5x
tg 8x
tg 10x
4tg 8x
tg (2-x)
tg (x+2)

f1 (х)
13 EMBED Equation.3 1415








f (х) = ctg x f1 (х) = - 13 EMBED Equation.3 1415
f (х)
1/5 ctg 5x
2 ctg 3x
ctg (x-1)
ctg 2x
ctg (5x-1)
5ctg 5x

f1 (х)
-13 EMBED Equation.3 1415
·






f (х) = 13 EMBED Equation.3 1415 f1 (х) =13 EMBED Equation.3 1415
f (х)
313 EMBED Equation.3 1415
513 EMBED Equation.3 1415
813 EMBED Equation.3 1415
413 EMBED Equation.3 1415
1/213 EMBED Equation.3 1415
613 EMBED Equation.3 1415

f1 (х)
13 EMBED Equation.3 1415






f (х) = 13 EMBED Equation.3 1415 f1 (х) =-13 EMBED Equation.3 1415
f (х)
313 EMBED Equation.3 1415
2,513 EMBED Equation.3 1415
3,413 EMBED Equation.3 1415
613 EMBED Equation.3 1415
1/213 EMBED Equation.3 1415
513 EMBED Equation.3 1415

f1 (х)
-13 EMBED Equation.3 1415







f (х) = 13 EMBED Equation.3 1415 f1 (х)= 13 EMBED Equation.3 1415
f (х)
13 EMBED Equation.3 1415
313 EMBED Equation.3 1415
1,213 EMBED Equation.3 1415
813 EMBED Equation.3 1415
413 EMBED Equation.3 1415
613 EMBED Equation.3 1415

f1 (х)
13 EMBED Equation.3 1415






f (х) = 13 EMBED Equation.3 1415 f1 (х)= -13 EMBED Equation.3 1415
f (х)
13 EMBED Equation.3 1415
613 EMBED Equation.3 1415
1,813 EMBED Equation.3 1415
313 EMBED Equation.3 1415
413 EMBED Equation.3 1415
1213 EMBED Equation.3 1415

f1 (х)
-13 EMBED Equation.3 1415







Задания на отработку для каждой формулы уровня В
Используя формулы производных, заполните таблицы:
f (х) = kx f1 (х) = k
f (х)
1/8x
4,85x
6,4x+13 EMBED Equation.3 1415
1,3x+ е
1/6x + 5
4,5x +2,3
1,9x
1,4x

f1 (х)
1/8








f (х) = xn f1 (х) = 13 EMBED Equation.3 1415
f (х)
X5
X8
3,5X20
1/813 EMBED Equation.3 1415
39x4
13 EMBED Equation.3 1415
6x14
84x25

f1 (х)
5х4








f (х) = 13 EMBED Equation.3 1415 f1 (х) = 13 EMBED Equation.3 1415
f (х)
3/х
13 EMBED Equation.3 1415
13 EMBED Equation.3 1415
13 EMBED Equation.3 1415
13 EMBED Equation.3 1415
13 EMBED Equation.3 1415
13 EMBED Equation.3 1415
-13 EMBED Equation.3 1415

f1 (х)
13 EMBED Equation.3 1415








f (х) = 13 EMBED Equation.3 1415 f1 (х) = - 13 EMBED Equation.3 1415
f (х)
13 EMBED Equation.3 1415
13 EMBED Equation.3 1415
13 EMBED Equation.3 1415
13 EMBED Equation.3 1415
13 EMBED Equation.3 1415
13 EMBED Equation.3 1415

f1 (х)
- 13 EMBED Equation.3 1415






f (х) = 13 EMBED Equation.3 1415 f1 (х) = 13 EMBED Equation.3 1415
f (х)
13 EMBED Equation.3 1415
13 EMBED Equation.3 1415
13 EMBED Equation.3 1415
13 EMBED Equation.3 1415
13 EMBED Equation.3 1415
13 EMBED Equation.3 1415
13 EMBED Equation.3 1415
13 EMBED Equation.3 1415
13 EMBED Equation.3 1415

f1 (х)
-13 EMBED Equation.3 1415









f (х) = 13 EMBED Equation.3 1415 f1 (х) = 13 EMBED Equation.3 1415
f (х)
13 EMBED Equation.3 1415
13 EMBED Equation.3 1415
13 EMBED Equation.3 1415
13 EMBED Equation.3 1415
13 EMBED Equation.3 1415
13 EMBED Equation.3 1415

f1 (х)
13 EMBED Equation.3 1415






f (х) = 13 EMBED Equation.3 1415 f1 (х) = 13 EMBED Equation.3 1415
f (х)
13 EMBED Equation.3 1415
13 EMBED Equation.3 1415
913 EMBED Equation.3 1415
13 EMBED Equation.3 1415
413 EMBED Equation.3 1415
613 EMBED Equation.3 1415

f1 (х)
13 EMBED Equation.3 1415


13 EMBED Equation.3 141513 EMBED Equation.3 1415



f (х) = 13 EMBED Equation.3 1415 f1 (х) =13 EMBED Equation.3 1415
f (х)
13 EMBED Equation.3 1415
13 EMBED Equation.3 1415
13 EMBED Equation.3 1415
13 EMBED Equation.3 1415
13 EMBED Equation.3 1415
13 EMBED Equation.3 1415

f1 (х)
13 EMBED Equation.3 1415






f (х) = 13 EMBED Equation.3 1415 f1 (х) = 13 EMBED Equation.3 1415
f (х)
13 EMBED Equation.3 1415
13 EMBED Equation.3 1415
13 EMBED Equation.3 1415
13 EMBED Equation.3 1415
13 EMBED Equation.3 1415
13 EMBED Equation.3 1415

f1 (х)
13 EMBED Equation.3 1415






f (х) = 13 EMBED Equation.3 1415 f1 (х) = 13 EMBED Equation.3 1415
f (х)
13 EMBED Equation.3 1415
13 EMBED Equation.3 1415
13 EMBED Equation.3 1415
13 EMBED Equation.3 1415
13 EMBED Equation.3 1415
13 EMBED Equation.3 1415

f1 (х)
13 EMBED Equation.3 1415






f (х) = 13 EMBED Equation.3 1415 f1 (х) = 13 EMBED Equation.3 1415
f (х)
13 EMBED Equation.3 1415
13 EMBED Equation.3 1415
13 EMBED Equation.3 1415
13 EMBED Equation.3 1415
13 EMBED Equation.3 1415+е
13 EMBED Equation.3 1415+П

f1 (х)
13 EMBED Equation.3 1415









f (х) = 13 EMBED Equation.3 1415 f1 (х) =13 EMBED Equation.3 1415
f (х)
13 EMBED Equation.3 1415
13 EMBED Equation.3 1415+ П
13 EMBED Equation.3 1415+е
1,13 EMBED Equation.3 1415
13 EMBED Equation.3 1415
13 EMBED Equation.3 1415

f1 (х)
13 EMBED Equation.3 1415

13 EMBED Equation.3 1415




f (х) = sin x f1 (х) = cos x
f (х)
3sin5 x
0.5sin8 x
sin13 EMBED Equation.3 1415
20sin 13 EMBED Equation.3 1415
5sin(3- x)
5sin 13 EMBED Equation.3 1415

f1 (х)
15cos x

13 EMBED Equation.3 1415cos 13 EMBED Equation.3 1415




f (х) = cos x f1 (х) = - sin x
f (х)
cos 3x
2,5cos 5x
13 EMBED Equation.3 1415cos 13 EMBED Equation.3 1415
4cos 20x
9cos13 EMBED Equation.3 1415
2Cos50 x

f1 (х)
-3sin3 x

-13 EMBED Equation.3 1415sin13 EMBED Equation.3 1415




f (х) = tg x f1 (х) = 13 EMBED Equation.3 1415
f (х)
tg 5x
6tg 8x
4,5tg 10x
4tg 8x
8tg (2-x)
3tg (x+2)

f1 (х)
13 EMBED Equation.3 1415






f (х) = ctg x f1 (х) = - 13 EMBED Equation.3 1415
f (х)
1/5 ctg 5x
2,5 ctg 3x
4ctg (x-1)
3ctg 2x
2ctg (5x-1)
20ctg 5x

f1 (х)
-13 EMBED Equation.3 1415






f (х) = 13 EMBED Equation.3 1415 f1 (х) =13 EMBED Equation.3 1415
f (х)
313 EMBED Equation.3 1415
513 EMBED Equation.3 1415
813 EMBED Equation.3 1415
13 EMBED Equation.3 1415
213 EMBED Equation.3 1415
613 EMBED Equation.3 1415

f1 (х)
13 EMBED Equation.3 1415






f (х) = 13 EMBED Equation.3 1415 f1 (х) =-13 EMBED Equation.3 1415
f (х)
313 EMBED Equation.3 1415
2,513 EMBED Equation.3 1415
3,413 EMBED Equation.3 1415
613 EMBED Equation.3 1415
1/213 EMBED Equation.3 1415
513 EMBED Equation.3 1415

f1 (х)
-13 EMBED Equation.3 1415














Для отработки навыков вычисления производных предлогаются тесты на соответствие.

ВАРИАНТ 1

f (х)
f1 (х)

1) cos(5-3x)
3 sin(5-3x)

2) 2ctgх
13 EMBED Equation.3 1415

3) f(x)= 13 EMBED Equation.3 1415
13 EMBED Equation.3 1415

4) ctg 1/х
13 EMBED Equation.3 1415

5) 13 EMBED Equation.3 1415
13 EMBED Equation.3 1415


ВАРИАНТ 2

f (х)
f1 (х)


f(x) = ctg(2x 2 -13 EMBED Equation.3 1415)
13 EMBED Equation.3 1415


f(x) = cosx +sinx + П
cosx - sinx


У= 13 EMBED Equation.3 1415
13 EMBED Equation.3 1415


f(x)=13 EMBED Equation.3 1415
13 EMBED Equation.3 1415


f(x) =(2х *sin 13 EMBED Equation.3 1415 + 1)2
2(х+1)

ВАРИАНТ 3

f (х)
f1 (х)

f(x) =2х*sinx,
2х(sin х*ln2+ cosx)


13 EMBED Equation.3 1415
13 EMBED Equation.3 1415


13 EMBED Equation.3 1415
13 EMBED Equation.3 1415

f(x)=13 EMBED Equation.3 1415

13 EMBED Equation.3 1415


f(x) = tgx+ctgx
13 EMBED Equation.3 1415




ВАРИАНТ 4

f (х)
f1 (х)


f(x) =13 EMBED Equation.3 1415НАЙТИ13 EMBED Equation.3 1415
4

Найдите значение производной функции: у(х)= tg(x) при х=
·/3
4

13EMBED Equation.31415
13EMBED Equation.31415


·(х)=(3х-4)ln(3х-4)
3ln(1+ln(3х-4))

f(x)=13 EMBED Equation.3 1415
13 EMBED Equation.3 1415



ВАРИАНТ 5

f (х)
f1 (х)


13EMBED Equation.31415
13EMBED Equation.31415

f(x)=13 EMBED Equation.3 1415
(2x+1) 13 EMBED Equation.3 1415

13EMBED Equation.31415
1/(x
·ln3)

13EMBED Equation.31415
13EMBED Equation.31415


13EMBED Equation.31415
13EMBED Equation.31415





1
2
3
4
5

Вариант 1






Вариант 2






Вариант 3






Вариант 4






Вариант 5




































































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