Partial derivative of integral with variable limit

In summary, the conversation discusses the use of Lagrange's formula to find the partial derivatives of a function G(\theta, k) and how the derivative with respect to \theta and k can be simplified using the fundamental theorem of calculus. The summary also mentions that the derivative with respect to \theta is just g(\theta, k) and the derivative with respect to k is \int_0^\theta \frac{\partial f}{\partial k} dx.
  • #1
Gregg
459
0

Homework Statement



[tex] G(\theta, k ) = \int^{\theta}_0 g(x,k) dx [/tex]

[tex] \frac{\partial G}{\partial \theta} = ? [/tex]

[tex] \frac{\partial G}{\partial k} = ? [/tex]

The Attempt at a Solution



If I say that [tex] \int g(x,k) dx = H(x,k) [/tex]

[tex] \int^{\theta}_0 g(x,k) dx = H(\theta,k) - H(0,k) [/tex]


Then is [tex]\frac{\partial G}{\partial \theta} = \frac{\partial H(\theta,k)}{\partial \theta}=g(\theta,k)[/tex]
?
 
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  • #2
Yup.
 
  • #3
That is "correct" but your derivation is more complicated than necessary! By the fundamental theorem of calculus, the derivative with respect to [itex]\theta[/itex] is just [itex]g(\theta, k)[/itex].

Lagrange's formula is an extension of the fundamental theorem of calculus:
[tex]\frac{d}{dx}\int_{\alpha(x)}^{\beta(x)} f(x, t) dt= f(x, \beta(x))}\frac{d\beta}{dx}- f(x, \alpha(x))\frac{d\alpha}{dx}+ \int_{\alpha(x)}^{\beta(x)} \frac{\partial f}{\partial x}dt[/tex].

Since [itex]\theta[/itex] only appears in the upper limit of the integral, the derivative with respect to [itex]\theta[/itex] is just
[tex]f(\theta, k)\frac{d\theta}{d\theta}= f(\theta, k)[/tex]

and the derivative with respect to k is just
[tex]\int_0^\theta \frac{\partial f}{\partial k} dx[/tex]
 
  • #4
That's clearer now, thank you.
 

1. What is the partial derivative of an integral with variable limit?

The partial derivative of an integral with variable limit is a mathematical concept that represents the rate of change of the integral with respect to one of its variables.

2. How is the partial derivative of an integral with variable limit calculated?

The partial derivative of an integral with variable limit is typically calculated using the fundamental theorem of calculus, which states that the derivative of an integral is equal to the function being integrated.

3. What is the significance of the partial derivative of an integral with variable limit?

The partial derivative of an integral with variable limit is important in many fields of science and engineering, as it allows for the analysis of how a system changes over time or in response to different inputs.

4. Can the partial derivative of an integral with variable limit be negative?

Yes, the partial derivative of an integral with variable limit can be negative, as it represents the direction and magnitude of change in the integral. A negative value indicates a decrease in the integral, while a positive value indicates an increase.

5. Is there a difference between a partial derivative and a total derivative of an integral with variable limit?

Yes, there is a difference between a partial derivative and a total derivative of an integral with variable limit. A partial derivative only considers the change in one variable, while a total derivative takes into account the change in all variables.

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