Is there a way to eliminate the second derivative in calculus using integration?

In summary, one way to eliminate the second derivative in the equation D²f(x) = g(x) is by integrating both sides to get ∫∫D²f(x)dx² = ∫∫g(x)dx². However, this method may not always work and it is important to understand the properties of differential operators before using them in equations.
  • #1
Jhenrique
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4
Given that D²f(x) = g(x), one form that eliminate the second derivate is integrating the equation: ∫∫D²f(x)dx² = ∫∫g(x)dx². But, and if I try so:

[tex]\\ \sqrt{D^2f(x)}=\sqrt{g(x)} \\ D\sqrt{f(x)}=\sqrt{g(x)} \\ PD\sqrt{f(x)}=P\sqrt{g(x)} \\ \sqrt{f(x)}=P\sqrt{g(x)} \\ f(x)=[P\sqrt{g(x)}]^2 \\ f(x)=[\int \sqrt{g(x)}dx]^2[/tex]

Is it works?
 
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  • #2
Going from the first to the second line you are claiming
[tex] \sqrt{ f''(x)} = \frac{d}{dx} \sqrt{f(x)} [/tex]

If we pick a random function, say f(x) = x4, the left hand side of this is
[tex] \sqrt{12 x^2} = \sqrt{12} |x| [/tex]
and the right hand side is
[tex] \frac{d}{dx} x^2 = 2x [/tex]

so we see they're not equal at all. The D operator is nice for seeing how you are using the linearity of the derivative but don't confuse it for an honest to goodness number that can be manipulated the same in every way.
 
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  • #3
Jhenrique said:
∫∫D²f(x)dx² = ∫∫g(x)dx².

You can't write ##dx^2## like that. It means integrate with respect to ##x^2## instead of integrating with respect to ##x##.
 
  • #4
Every definition of the "differential operators" will include showing which ordinary arithmetic operators work for differential operators as well. Of course you can only use arithmetic operators that have the property that they can be applied to differential operators. Have you seen any proof that the "square root" operator has that property?
 
  • #5
Not always there is demonstrations for certain relations. I never saw a demo for this but I know it's valid.

[tex]\\ f^D=f^D \\ log(f^D)=log(f^D) \\ log(f^D)=Dlog(f) \\ f^D=exp(\frac{f'}{f})[/tex]
PS: f^D = f*(x) = geometric derivate
 

1. What is the "D operator" in calculus?

The "D operator" in calculus refers to the derivative operator, denoted as d/dx. It is used to represent the rate of change or slope of a function at a specific point.

2. How is the "D operator" used in calculus?

The "D operator" is used to find the derivative of a function with respect to the independent variable. It is also used in differential equations to represent the rate of change of a dependent variable with respect to an independent variable.

3. What are the properties of the "D operator"?

The "D operator" has several properties, including linearity, product rule, quotient rule, and chain rule. These properties allow for the simplification of complex expressions and the calculation of derivatives for various types of functions.

4. How is the "D operator" related to integration?

The "D operator" and integration are inverse operations. The integral of a function is the anti-derivative, or the function whose derivative is the original function. This relationship is represented by the fundamental theorem of calculus.

5. What are some real-world applications of the "D operator" in calculus?

The "D operator" is used in many fields, including physics, engineering, economics, and statistics. It is used to model and analyze various phenomena, such as motion, growth, and change. Examples include calculating velocity, finding maximum and minimum values, and optimizing functions.

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