Proof of Antiderivative: Techniques & Math Behind x^n

In summary, there are several techniques for derivation, such as the method of incrementation and working with limits, that have shown the derivative of x^n to be n*x^(n-1). There is also the Fundamental Theorem of Calculus, which proves that integration corresponds to antidifferentiation. The proof for this theorem is called the "Fundamental Theorem of Calculus". Integration is considered an "inverse" problem, where we are given a formula and must solve for the variable, while differentiation is a "direct" problem, where we are given a formula and must evaluate it. The anti-derivatives of most integrable functions cannot be given in terms of elementary functions, but we can find a function whose derivative is the given function
  • #1
Gear300
1,213
9
There are several techniques for derivation that have proved that the derivative for x^n is n*x^(n-1). There is the method of incrementation, working extensively with limits, etc... I'm unsure of the antiderivative. Is there any mathematical proof for the antiderivative or did they simply say that it reverses differentiation and came up with it just by looking at the method for derivation?
 
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  • #2
well by definition:

[tex]\int x^N dx =\frac{x^{N+1}}{N+1} + C[/tex]

so you can just use that definition.
 
  • #3
rock.freak667 said:
well by definition:

[tex]\int x^N dx =\frac{x^{N+1}}{N+1} + C[/tex]

so you can just use that definition.

That's not a definition. It's a theorem. The proof that integration corresponds to antidifferentiation is called the "Fundamental Theorem of Calculus".
 
  • #4
ah well I learned something today
 
  • #5
differentiate it!
 
  • #6
Throughout mathematics we come upon "direct" problems and "indirect" or "inverse" problems. Here's a simple algebraic example: if you are given the function f(x)= [itex]x^5- 7x^4+ 3x^3- 2x^2+ x- 5[/itex] and asked "what is f(1)?", that's easy. You are given the formula to use and just do the arithmetic: f(1)= 1- 7+ 3- 2+ 1- 5= -9. If, however, you are asked to solve the equation, f(x)= [itex]x^2- 7x^4+ 3x^3- 2x^2+ x- 5= -9[/itex], that's extremely difficult! There is no "formula" for solving general 5th degree equations. Here, because we had already done the evaluation, we know that one solution is x= 1. However, there may be up to 4 more solutions that we don't know.

Differentiation is a "direct" problem because we are given a formula for it. Integration, or "anti-differentiation" is defined only as the inverse of differentiation. A result of that fact is that the anti-derivatives of most integrable functions cannot be given in terms of elementary functions.

If we can "remember" or otherwise find a function whose derivative is the given function- for example, we know that the derivative of [itex](1/(n+1)) x^{n+1}[/itex] is [itex]x^n[/itex]- then we know that the new function is the "anti-derivative" of the original function- [itex]x^n[/itex] is the anti-derivative of [itex](1/(n+1))x^{n+1}[/itex].
Of course, we also know that there are an infinite number of other functions having that same derivative. fortunately, we can prove, using the mean value theorem, that if two functions have the same derivative, they must differ only by a constant- we know that any anti-derivative of [itex]x^n[/itex] must be of the form [itex](1/(n+1))x^{n+1}+ C[/itex] where C is a constant.
 
  • #7
I see. A lot of information given, thanks.
 
  • #8
Gear300 said:
Is there any mathematical proof for the antiderivative or did they simply say that it reverses differentiation and came up with it just by looking at the method for derivation?

When you say "anti derivative" it implies, the process of differentiation reversed. From the rest of your sentence I think you may have instead wanted to use the word "integral". The Only reason we can use those words interchangeably is because of our knowledge of the Fundamental theorem of Calculus, but remember they were originally two different things.

Now if that quote was asking if there is another way to prove [tex]\int^b_a x^n = \frac{b^{n+1} - a^{n+1}}{n+1}[/tex], then yes. Riemann sums.
 

1. What is a proof of antiderivative?

A proof of antiderivative is a mathematical process used to determine the original function from its derivative. In other words, it is a way to find the function that, when differentiated, will give the given derivative. This is useful in many areas of mathematics, including calculus and physics.

2. What techniques are used in a proof of antiderivative?

There are several techniques that can be used to prove the antiderivative of a function. Some of the most commonly used techniques include the power rule, substitution, integration by parts, and partial fractions. Each of these techniques involves a different approach to finding the antiderivative, and they can be used in combination to solve more complex problems.

3. How is the power rule used in a proof of antiderivative?

The power rule is a simple and commonly used technique in a proof of antiderivative. It states that the antiderivative of a function with the form x^n is (x^(n+1))/(n+1) + C, where C is the constant of integration. This rule can be extended to more complex functions by using algebraic manipulation and the chain rule.

4. Can substitution be used in a proof of antiderivative?

Yes, substitution is a powerful technique in a proof of antiderivative. It involves replacing a variable in the integral with a new variable, which simplifies the problem and makes it easier to solve. Substitution is particularly useful when dealing with trigonometric functions and exponential functions.

5. What is the importance of proof of antiderivative in calculus?

The proof of antiderivative is crucial in calculus because it allows us to find the original function from its derivative. This is essential in solving many real-world problems, such as finding the distance traveled by an object given its velocity or determining the change in temperature over time. Additionally, the concept of antiderivatives is fundamental in the study of integrals and the fundamental theorem of calculus.

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