Properties of Differential Operator and Proper Formalism

In summary: So it is basically the derivative of the derivative, which is the second derivative.In summary, the conversation discusses the use of the operator d/dx to differentiate a function defined by y=cx. The chain rule/implicit differentiation is used on the left while normal differentiation is used on the right. The proper usage of the formalism for finding the second derivative is also explained. The second derivative is represented by the notation d^2y/dx^2, which is the derivative of the derivative.
  • #1
Nano-Passion
1,291
0
Let us take a function defined by

[tex]y=cx[/tex]
To differentiate that, we use the operator d/dx
[tex]\frac{d}{dx} y = \frac{d}{dx} cx^{-1}[/tex]
By the chain rule/implicit differentiation on the left and normal differentiation on the right we get,
[tex]\frac{dy}{dx} = -1c x^{-2}[/tex]

What confuses me is the proper usage of the formalism to get

[tex]\frac{d^2y}{dx^2} = 2cx^{-3}[/tex]

It doesn't seem that we can use the same operation as last time.
 
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  • #2
Your initial calculation is wrong, x is not supposed to be x^(-1), it remains x, and after the differentiation it is 1. The second derivative is then, of course, 0.
 
  • #3
Why not?

##\dfrac{d^2y}{dx^2}## is just a (slightly confusing, if you think too hard about what ##dx^2## is supposed to mean) abbreviation for ##\dfrac{d}{dx}\left(\dfrac{d}{dx}y\right)##.

(I assumed your first equation was a typo for ## y = cx^{-1}##).
 
  • #4
meldraft said:
Your initial calculation is wrong, x is not supposed to be x^(-1), it remains x, and after the differentiation it is 1. The second derivative is then, of course, 0.
Typo.
AlephZero said:
Why not?

##\dfrac{d^2y}{dx^2}## is just a (slightly confusing, if you think too hard about what ##dx^2## is supposed to mean) abbreviation for ##\dfrac{d}{dx}\left(\dfrac{d}{dx}y\right)##.

(I assumed your first equation was a typo for ## y = cx^{-1}##).

Oh thank you! Now I realize why it is written in that form
 
  • #5


I can understand your confusion about the usage of formalism in differentiating a function. The differential operator, represented by d/dx, is a mathematical tool that helps us find the rate of change of a function with respect to its independent variable. In this case, the function y=cx is a simple linear function, and its first derivative can be easily found using the operator d/dx.

However, when it comes to finding the second derivative, we need to use a different approach. The proper formalism for finding the second derivative of a function is to use the operator d^2/dx^2. This operator represents the second derivative with respect to the independent variable x.

In the case of y=cx, the second derivative can be found by applying the operator d^2/dx^2 to the first derivative \frac{dy}{dx} = -1c x^{-2}. This results in \frac{d^2y}{dx^2} = 2cx^{-3}, as you have correctly mentioned.

It is important to understand that the usage of formalism in differentiating a function depends on the complexity of the function and the degree of the derivative being calculated. In more complex functions, we may need to use higher-order operators such as d^3/dx^3 or even d^4/dx^4 to find the third and fourth derivatives, respectively.

In summary, the properties of the differential operator and proper formalism are essential tools in mathematics and science, and their proper usage helps us accurately calculate derivatives of functions. I hope this explanation has helped clear your confusion.
 

1. What is a differential operator?

A differential operator is a mathematical operator that acts on a function to produce another function. It is typically represented by a symbol, such as ∂ or d, and involves differentiation with respect to one or more variables.

2. What are the properties of a differential operator?

The properties of a differential operator include linearity, commutativity, and associativity. Linearity means that the operator distributes over addition and scalar multiplication. Commutativity means that the order in which the operator acts on functions does not affect the result. Associativity means that the order in which multiple operators act on a function does not affect the result.

3. How is formalism used in differential operators?

Formalism is a mathematical approach that uses symbols and rules to represent and manipulate abstract concepts and operations. In differential operators, formalism is used to express the properties and actions of a differential operator in a concise and systematic way.

4. What is the proper formalism for differential operators?

The proper formalism for differential operators is known as the "graded commutative algebra" or "exterior algebra." This formalism uses symbolic notation to represent the properties and actions of differential operators, making it easier to perform calculations and prove theorems.

5. How are differential operators used in science?

Differential operators are used in many areas of science, particularly in physics and engineering. They are used to describe and solve mathematical models of physical phenomena, such as heat transfer, fluid flow, and quantum mechanics. Differential operators also play a crucial role in the development of mathematical tools and techniques for analyzing and understanding complex systems.

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