Derivative of an Integral

In summary, the conversation discusses the validity of switching the order of differentiation and integration, with the conclusion that this is indeed a valid operation. The conversation also touches on the difference between the ##dx## in integral notation and the ##dx## of differentials.
  • #1
Hertz
180
8
Hi, so this is just a quick question about taking a derivative of an integral. Assume that I have some function of position ##A(x, y, z)##, then assume I am trying to simplify $$D_i\int{A dx_j}$$ where ##i≠j##. So, I'm taking the partial derivative of the integral of A, but the derivative and integral are with respect to different variables.

Considering that the integral is similar to a summation, I intuitively believe that I can take this step:
$$D_i\int{A dx_j}=\int{D_i(A dx_j)}$$

This is where I am confused. Can I take this step?:
$$\int{D_i(A dx_j)}=\int{D_i(A) dx_j}$$

I believe that this is right, but I don't exactly know why. It's as if ##dx_j## is a constant. Is this the case? I can't really see why on my own because I've seen instances where differentials depend on other differentials and obviously I've seen cases where one differential over another differential changes with position, which implies the differentials themselves change with respect to each other. I don't know, I'm just confused about how to think about this. How can I justify taking the ##dx_j## out of the ##D_i##?

Lastly, what about the reverse case, where I have ##\int{D_i(A) dx_j}## can I convert this too ##D_i\int{A dx_j}##? Thanks!
 
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  • #2
The short answer is, yes - switching the order of differentiation/integration in the manner you have indicated is valid. The long answer is https://en.wikipedia.org/wiki/Leibniz_integral_rule.

Also be aware that the ##dx## in the integral notation, at least as far the Reimann integral is concerned, is not the same as the ##dx## of differentials. The integral version is more of a notational artifact that indicates the variable of integration. It's not uncommon to see the two versions mix and mingle, such as when studying differential equations or setting up physics problems and other applications problems. and most of the time it's okay because it works. But just be aware that there's a minor abuse of notation going on in those cases.
 

1. What is the derivative of an integral?

The derivative of an integral is a mathematical operation that finds the rate of change of the integral function with respect to its independent variable. In other words, it measures how fast the integral function is changing at a specific point.

2. Why is the derivative of an integral important?

The derivative of an integral is important because it allows us to find the slope of a curve at a specific point, which is useful in many real-world applications such as physics, engineering, and economics. It also helps us to solve optimization problems and to understand the behavior of a function.

3. How do you find the derivative of an integral?

The derivative of an integral can be found using the fundamental theorem of calculus, which states that the derivative of an integral is equal to the integrand function evaluated at the upper limit of integration. This means that to find the derivative, you simply need to plug in the upper limit of integration into the integrand function and simplify.

4. Can the derivative of an integral be negative?

Yes, the derivative of an integral can be negative. This indicates that the integral function is decreasing at that particular point.

5. How is the derivative of an integral related to the area under a curve?

The derivative of an integral is related to the area under a curve by the fundamental theorem of calculus. The derivative of the integral function is equal to the height of the curve at a specific point, which is also the slope of the tangent line to the curve at that point. This means that the derivative represents the rate of change of the area under the curve, which is equal to the height of the curve at that point.

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