# Questions tagged [leibniz-integral-rule]

Also known as Feynman's trick or differentiation under the integral sign.

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### Derivative of double(multiple) integrals

How do I find the derivative of function f with respect to t: $$f(t) = \int_{ }^{ }\int_{ }^{ }\sqrt{x^{2}+y^{2}}dxdy$$ and the domain of integral is: $$(x-t)^2 + (y-t)^2 \leq 1$$
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### Reasonable and Logical Integration

$\lim _{k\to +\infty}$ $\int_{0}^{k[x]} (kt-[kt])^k dt$ $;$ $k\in N$ is $[\frac{\lambda x} 2]$ Where $[.]$ denotes greatest Integer function then the value of $\lambda$ would be? My approach to this ...
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### Two-dimensional generalization of Leibniz's integral rule

Given a function $f(x,y):\mathbb R\times \mathbb R\to \mathbb C$ and a real parameter $\theta$, one can use Leibniz's integral rule to solve \label{eq}\tag{1}\frac{d}{d\theta}\int_{a(\...
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### How to Calculate the Partial Derivative F_u(1,v) of the Integral Function F(u,v)

I'm working on a problem in calculus and am having difficulty with a specific function and its partial derivative. The function is defined as: $F(u,v) = \int_{uv}^{u+v}e^{-(u-y)^2}dy$ I'm trying to ...
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### Help with Variational Calculus & Leibnitz Rule

Let $f: (0,\infty)\to \mathbb{R}$ be convex and lower-semicontinuous with $f(1)=0$ and $\mu$, $\hat{\mu}$ be two probability distributions on a measurable space $\mathcal{X}$ which are absolutely ...
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### How to apply Leibniz rule for the derivation of an integral when the integration variable depends on the parameter

I know that, according to the statement of the Leibniz rule for the derivation of a parametric integral, if we have: $$F(\lambda)=\int^{b(\lambda)}_{a(\lambda)}f(\lambda,x)dx$$ then the following ...
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### Time derivative of integrals on changing level and upper contour sets

There are related questions and answers on StackExchange but I cannot find a general formula for the following types of (time) derivatives of integrals with respect to changing level and upper contour ...
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### The integral $I(a,b)=\int_0^1x^a\log^b(x) \space dx$

I found a formula for the integral $$I(a,b)=\int_0^1x^a\log^b(x) \space dx$$ By just differentiating the integral $I(t)=\int_0^1x^t\space dx$ under the integral sign b times and evaluating the ...
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### How is Leibniz integration rule applied to stochastic integrals? [duplicate]

I am working through the mathematics behind the Hull-White short rate model and am currently stuck on how to take the partial derivative of and evaluate a stochastic integral when looking at how bond ...
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### Applying Leibniz rule for double integral

I'm trying to solve A = $\frac{d}{dy} \int_{0}^{y} \int_{0}^{k(x_1,y)} (g(x_1)+h(x_2))f(x_2)dx_2 f(x_1)dx_1$. Here is my approach: let $j(x_1, y) = \int_{0}^{k(x_1,y)} (g(x_1)+h(x_2))f(x_2)dx_2$, then ...
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### Use Leibnitz Theorem to solve the following- [closed]

If anyone could break it down to smaller steps, I'd appreciate alot.
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### Leibniz integral rule for the Airy function.

We recall that the Airy function is (well) defined by: $\text{Ai}(x) = \frac{1}{\pi} \int_0^\infty \cos\left(\frac{t^3}{3} + xt\right) dt$ I want to apply the theorem of differentiation under the ...
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### The Leibniz integral rule

I've had problems applying the Leibniz Rule. The professor's slides say: Consider the following value function \nu(t) = \int_{t}^{\infty} e^{- \int_{t}^{s}r(z)dz} \pi(s) ds \end{...
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### A proof of Leibniz integral rule

I'm reading Leibniz integral rule Let $X$ be an open subset of $\mathbb R^d$ and $(\Omega, \mathcal F, \mu)$ a measure space. Let $f:X \times \Omega \to \mathbb R, (x, \omega) \mapsto f(x, \omega)$. ...
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### Is the Local Average of a Continuous Multivariable Function Differentiable?

Suppose we have a continuous $f:\mathbb{R}\to\mathbb{R}$. It is an immediate corollary of the Leibniz integral rule that $f^*:x\mapsto\int_{x-\frac{1}{2}}^{x+\frac{1}{2}}f(t)\ dt$, the "local&...
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### Leibniz Integral Rule applied n-times

I was trying to come up with a generalization of the Leibniz Integral Rule $$\frac{d}{dx}\left(\int_a^xf(x,t) \; dt\right) = f(x,x) + \int_a^x \frac{\partial}{\partial x} f(x,t) \; dt$$ and ...
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### Reynolds transport theorem: link between different formulations

Currently I am studying the reynolds transport theorem. From my research I could find two depictions of the theorem. My problem lies withing tying these two formulations together. "Mathematical/...
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### Fundamental theorem of calculus with the derivative on the inside?

I know that: $$\frac{d}{dx}\int_{a}^{g(x)} f(t)dt = f(g(x))*g'(x)$$ But what about: $$\int_{a}^{g(x)}\frac{d}{dx}f(x)dx$$ An example of this would be: $$\int_{3}^{t^3}\frac{d}{dx}\frac{x}{x-2}dx$$ Do ...
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### Leibniz formula for integral with spherical coordinates in multi-dimensional space

Given $u: \mathbb{R}^n \to \mathbb{R}$ smooth enough and set $\bar{u}(r) = \int_{|x| = r}u(x)d\sigma$, with $x = \left(r, \sigma\right)$ is the spherical coordinate of $x$. I have some differences in ...
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### the Bayes rule with a general form (Young textbook problem 3.3)

Hi so I am trying to find the bayes rule with a loss function defined as for a general estimator. we just need to find the general form of the Bayes rule for this loss function. let the prior be ...
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### Can we prove the following by using Leibniz's rule for differentiation of a definite integral?

Let $\Phi(t)$ be $$\Phi(t)=\int\limits_t^\infty \kappa e^{-\int\limits_t^s g(\omega)d\omega} ds$$ such that $0<t<s$. If it holds that $\Phi(t)=\alpha$ where $\alpha$ is constant $\forall t$ can ...
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### How to find the derivative w.r.t. lower limits?

How to find $\frac{d}{dy} \int_{y}^{\infty} \int_{2y}^{\infty} y f(x_2)dx_2 f(x_1)dx_1$, where $x_1$ and $x_2$ are two independent continuous random variables and $f(x_1)$ and $f(x_2)$ are their PDFs. ...
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