Derivative of Integral Function with Respect to a Space-Time Variable

In summary, the problem involves finding the derivative of an integral function, where \theta is a function of \xi and \tau (space and time variables). The goal is to eliminate the integral by differentiating the equation with respect to \xi once, using the product method of differentiation. However, the integral is still present in the final equation. Assistance is needed in eliminating the integral.
  • #1
saravanan13
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0

Homework Statement


Derivative of an integral function. [tex]\theta[/tex] is a function [tex]\xi[/tex] and [tex]\tau[/tex](space and time variable)


Homework Equations


cos[tex]\theta[/tex][tex]\frac{\partial}{\partial[tex]\tau[/tex]}[/tex]([tex]\int[/tex]
sin[tex]\theta[/tex]d[tex]\xi[/tex]).
The limit running from -[tex]\infty[/tex] to [tex]\xi[/tex]
[tex]\theta[/tex][tex]\rightarrow[/tex] zero as [tex]\xi[/tex][tex]\rightarrow[/tex] to -[tex]\infty[/tex]
I want to eliminate the integral by differentiating the above equation with respect to [tex]\xi[/tex] once.

The Attempt at a Solution


I simply differentiated with respect to [tex]\xi[/tex] by product method of differentiation.
but still integral retained in the final equation
This problem i encountered in journals.
 

Attachments

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  • #2
\frac{\partial}{\partial\xi}[cos\theta\frac{\partial}{\partial\tau}(\intsin\thetad\xi)]=sin\theta\frac{\partial\theta}{\partial\xi}\frac{\partial}{\partial\tau}(\intsin\thetad\xi)+cos\theta\frac{\partial}{\partial\tau}(\frac{\partial}{\partial\xi}(\intsin\thetad\xi))But still integral retained. Please help me out to eliminate the integral.
 

1. What is the derivative of an integral?

The derivative of an integral is the original function that was integrated. It represents the rate of change of the original function at a specific point.

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

To find the derivative of an integral, you can use the Fundamental Theorem of Calculus, which states that the derivative of an integral is the original function. You can also use the chain rule or other differentiation rules depending on the complexity of the integral.

3. Can the derivative of an integral be negative?

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

4. What is the relationship between the derivative and integral?

The derivative and integral are inverse operations. The derivative of an integral is the original function, and the integral of a derivative is the original function plus a constant. They are also connected through the Fundamental Theorem of Calculus.

5. Why is the derivative of an integral important?

The derivative of an integral is important because it allows us to find the instantaneous rate of change of a function at a specific point. This has various applications in physics, engineering, and other fields where understanding the rate of change is crucial.

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