Leibniz Rule (derivative of an integral)

In summary, the Leibniz rule states that if a continuous two variable function f is defined on a closed interval and its partial derivative with respect to the first variable exists and is continuous on the same subset, then the derivative of the integral of f with respect to the second variable exists and is equal to the integral of the partial derivative of f with respect to the second variable. This can be proven using the Mean Value Theorem or a simpler proof that utilizes uniform continuity. Riemann integrability and limits can be exchanged if everything exists and the limit is uniformly continuous in the second variable.
  • #1
Castilla
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The Leibniz rule:

1. Let f(x,y) be a continuous two variable real function defined on (closed intervals) {x0, x1} x {y0, y1}.
2. Let f_1 (partial derivative of f with respect to the first variable) exists and be continuous on the same subset of RxR.
3. Let F be defined as F(x) = (int) (lim x = y0, x = y1) f(x,y)dy

Then F' exists and F'(x) = (int) (lim x = y0, x = y1) f_1(x,y)dy.

I know and understand the proof that uses the Mean Value Theorem of elementary calculus. Yet in Wikipedia (Leibniz's rule: derivatives and integrals) they use a simpler proof. They arrived to this point, which I understood:

F'(y) = lim (h->0) (int)(lim x=c, x=d) ( f(x+h, y) - f(x,y)) dy
h

BUt then they say: "... and using uniform continuity the right hand side equals to

(int) (lim x=c, x=d) f_1(x,y)dy."

How do they manage to use uniform continuity to introduce the limit inside the integral ?

Thanks.
 
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  • #2

1. What is the Leibniz Rule?

The Leibniz Rule, also known as the Fundamental Theorem of Calculus, is a mathematical formula that relates the derivative of an integral to the original function.

2. How is the Leibniz Rule used?

The Leibniz Rule is used to solve problems where the function being integrated is dependent on another variable. It allows for the simplification of complex integrals by taking the derivative of the integral rather than the original function.

3. What is the formula for the Leibniz Rule?

The formula for the Leibniz Rule is d/dx ∫a^b f(x,t) dt = f(x,b) - f(x,a), where f(x,t) is the function being integrated and a and b are the limits of integration.

4. Can the Leibniz Rule be used for any type of integral?

No, the Leibniz Rule can only be used for definite integrals with fixed limits of integration. It cannot be used for indefinite integrals or integrals with varying limits.

5. How does the Leibniz Rule relate to the Mean Value Theorem?

The Leibniz Rule is a generalization of the Mean Value Theorem, which states that if a function is continuous on a closed interval and differentiable on an open interval within that closed interval, then there exists a point where the derivative of the function is equal to the slope of the secant line connecting the endpoints of the interval. The Leibniz Rule extends this concept to integrals, allowing for the calculation of the derivative of an integral.

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