Can Cross Derivatives be Equal in Partial Derivatives?

In summary, the conversation discusses a theoretical question about partial derivatives and cross derivatives of a function x(i,j). The question asks if it is logical for the cross derivatives to be equal, and the conversation includes a potential solution using a Lagrangian.
  • #1
chicago77
2
0
So this is a theoretical question I am not sure about with partial derivatives.

Say we have function x(i,j) with a=dx/di and b=dx/dj

Now is this logical when the cross derivatives are required to be equal?

db/da --> substitute b
= d^2x/(dadj) substitute da
= d^2x/((d^2x/di)dj)
=di/dj

Is this allowed?
 
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  • #2
chicago77 said:
So this is a theoretical question I am not sure about with partial derivatives.

Say we have function x(i,j) with a=dx/di and b=dx/dj

Now is this logical when the cross derivatives are required to be equal?

db/da --> substitute b
Right off the bat I'm not following this. This says you are taking the (partial) derivative with respect to dx/dj, which makes not sense at all.

A small part of my difficulty is with your notation - dx/di - which is supposed to be the partial derivative of x with respect to i.
chicago77 said:
= d^2x/(dadj) substitute da
= d^2x/((d^2x/di)dj)
=di/dj

Is this allowed?
Presumably i and j (which are really bad choices for the names of variables) are independent variables, so you wouldn't ordinarily take the derivative (partial or otherwise) of one with respect to the other.
 
  • #3
nvm I have solved this with a lagrangian.
 
Last edited:

What is the Second Derivative Rule?

The Second Derivative Rule is a mathematical rule used in calculus to find the rate of change or curvature of a function. It involves taking the derivative of the derivative of a given function.

When is the Second Derivative Rule used?

The Second Derivative Rule is used when analyzing the concavity and inflection points of a function. It helps determine whether the function is increasing or decreasing at a given point and the direction of its curvature.

How is the Second Derivative Rule applied?

To apply the Second Derivative Rule, first take the derivative of the given function to find its first derivative. Then, take the derivative of the first derivative to find the second derivative. The second derivative can then be evaluated at a specific point to determine the concavity and curvature of the function at that point.

What does a positive or negative second derivative indicate?

A positive second derivative indicates that the function is concave up at a given point, meaning it is increasing and curving upwards. A negative second derivative indicates that the function is concave down, meaning it is decreasing and curving downwards.

Why is the Second Derivative Rule important?

The Second Derivative Rule is important because it allows us to analyze the behavior of a function and its rate of change in a more precise manner. It helps us understand the curvature and inflection points of a function, which is useful in many real-world applications such as physics, economics, and engineering.

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