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Antiderivative Chain Rule Antiderivative Chain Rule Chain rule plays a very important role when we solve Integration, before going to chain rule, we should discuss about Antiderivative , it generally means summing up the things. It is widely used in calculating the area under the curve. For calculating the area we will take a very small segment as dx then we will take a initial limit and a final limit and we will integrate the given function between these limit and we can easily calculate the area under the given curve. Chain rule is only applied when we have two functions f(x) and g(x). We have seen that we can also solve two function problem with integration by substitution as well but doing it from chain rule is little easier. We will see an example how we can solve any given function with the help of chain rule In differential calculus, we often mention chain rule in finding the derivative of a function especially composite functions. Now, we'll discuss inverse chain rule method. Inverse chain rule is a method of finding antiderivatives or integrals of a function by guessing the integral of that function, and then differentiating back using the chain rule.

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With that being said, let me introduce to you the following theorem..So, if an integral is in the form reflected in left-side of the integral above, then we can easily find its derivative by doing inverse chain method. Also, most integrals like the one above can be also solved by doing substitution, that's why inverse chain rule anti-differentiation is also called as substitution method but a special case of substitution rule. Illustration In this illustration, let us solve one integral of a function through inverse chain rule. Say, we are to integrate 3x2(3+x3)8dx . Firstly, by looking at the integral, we can say that the integral, which is a product, can be integrated by simplifying the integrand into a polynomial. But you don't want to do that! Imagine a polynomial raised to eighth powers. That's a lot of terms. In this illustration, let us solve the integral of the above function through inverse chain rule. We can easily integrate such integral through inverse chain rule because it is in the form f(x)nf(x)dx . Why? , the term 3+x3 corresponds to f(x) and if we integrate 3+x3 the answer is 3x2dx which just corresponds to f(x)dx regardless of the constant. Thus, according to the theorem above, Example:The function F(x) = x3/3 is an antiderivative of f(x) = x2. As the derivative of a constant is zero, x2 wil have an infinite number of antiderivatives; such as (x3/3) + 0, (x3/3) + 7, (x3/3) - 42, (x3/3) + 293 etc. Thus, al the antiderivatives of x2 can be obtained by changing the value of C in F(x) = (x3/3) + C; where C is an arbitrary constant known as the constant of integration. Essential y, the

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graphs of antiderivatives of a given function are vertical translations of each other; each graph's location depending upon the value of C. In physics, the integration of acceleration yields velocity plus a constant. The constant is the initial velocity term that would be lost upon taking the derivative of velocity because the derivative of a constant term is zero. This same pattern applies to further integrations and derivatives of motion (position, velocity, acceleration, and so on).

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Antiderivative Chain Rule  

Inverse chain rule is a method of finding antiderivatives or integrals of a function by guessing the integral of that function, and then dif...

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