Производные элементарных функций.

Правила дифференцирования.

Если у функций f(x) и g(x) существуют производные, то

Производная сложной функции:

Геометрический смысл производной. Производная в точке x₀ равна угловому коэффициенту касательной к графику функции y=f(x) в этой точке

Уравнение касательной к графику функции y=f(x) в точке x₀:

Физический смысл производной.

Если точка движется вдоль оси х и ее координата изменяется по закону x(t), то мгновенная скорость точки:

Describe the differential of the function. Higher order derivatives.

Если функция имеет производную в каждой точке своей области определения, то ее производная есть функция от . Функция , в свою очередь, может иметь производную, которую называют производной второго порядка функции (или второй производной ) и обозначают символом . Таким образом

Найти производную второго порядка функции

Согласно определению, вторая производная - это первая производная от первой производной, то есть

Поэтому сначала найдем производную первого порядка от заданной функции согласно правилам дифференцирования и используя таблицу производных:

Теперь найдем производную от производной первого порядка. Это будет искомая производная второго порядка:

Ответ

Вычисления производной любого порядка, формула Лейбница

Для вычисления производной любого порядка от произведения двух функций, минуя последовательное применениеформулы вычисления производной от произведения двух функций, применяется формула Лейбница:

где ,

- факториал натурального числа

Задание. Найти , если

Решение. Так как заданная функция представляет собой произведение двух функций , , то для нахождения производной четвертого порядка целесообразно будет применить формулу Лейбница:

Найдем все производные и посчитаем коэффициенты при слагаемых.

1) Посчитаем коэффициенты при слагаемых:

2) Найдем производные от функции :

3) Найдем производные от функции :

Тогда

Ответ.

4. Describe Hospital Rues and show its using in finding limit of functions.

This is the problem with indeterminate forms. It’s just not clear what is happening in the limit. There are other types of indeterminate forms as well. Some other types are,

These all have competing interests or rules that tell us what should happen and it’s just not clear which, if any, of the interests or rules will win out. The topic of this section is how to deal with these kinds of limits.

As already pointed out we do know how to deal with some kinds of indeterminate forms already. For the two limits above we work them as follows.

In the first case we simply factored, canceled and took the limit and in the second case we factored out an from both the numerator and the denominator and took the limit. Notice as well that none of the competing interests or rules in these cases won out! That is often the case.

So we can deal with some of these. However what about the following two limits.

This first is a 0/0 indeterminate form, but we can’t factor this one. The second is an indeterminate form, but we can’t just factor an out of the numerator. So, nothing that we’ve got in our bag of tricks will work with these two limits.

L’Hospital’s Rule

Suppose that we have one of the following cases,

where a can be any real number, infinity or negative infinity. In these cases we have,

So, L’Hospital’s Rule tells us that if we have an indeterminate form 0/0 or all we need to do is differentiate the numerator and differentiate the denominator and then take the limit.

Solution

(a)

So, we have already established that this is a 0/0 indeterminate form so let’s just apply L’Hospital’s Rule.

(b)

In this case we also have a 0/0 indeterminate form and if we were really good at factoring we could factor the numerator and denominator, simplify and take the limit. However, that’s going to be more work than just using L’Hospital’s Rule.

(c)

This was the other limit that we started off looking at and we know that it’s the indeterminate form so let’s apply L’Hospital’s Rule.

Now we have a small problem. This new limit is also a indeterminate form. However, it’s not really a problem. We know how to deal with these kinds of limits. Just apply L’Hospital’s Rule.

Sometimes we will need to apply L’Hospital’s Rule more than once.

Explain double integral.

Integration is a very important part of calculus. There are many types of integrations like simple integration, double integration, triple integration. We will discuss double integration here. Double integration is mainly used to find the surface area of a 2d figure. We can easily find the area of a rectangular region by double integration. If we know simple integration, then it will be easy to solve double integration problems.

So, first of all, we will discuss some basic rules of integration. Then, we will discuss about the applications of those rules for double integration. There are many applications of double integration. We will discuss those below.

Rule for Double Integration by Parts:

Question: Find the area of the function bounded by

The bounded region of the given functions is below in the graph, in which we take vertical stripes to find the limits of the integration

Now, we can see that the lower part of the strip lie on y = x curve and upper part lie on

x = 0 t0 x = 1. Hence, we have the limits y = x to y = x√ x and x = 0 to 1. Then,

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