# Thread: derivative with x occurring at both integrand and upper bound

1. ## derivative with x occurring at both integrand and upper bound

If $f$ is continuously differentiable in $\mathbb R^2$, $g$ is continuously differentiable in $\mathbb R$, $a$ a constant real, how to calculate the derivative of $\int_a^{g(x)}f(x,t)dt$ with regard to $x$ (not by numerical methods)? Thanks.

2. Originally Posted by zzzhhh
If $f$ is continuously differentiable in $\mathbb R^2$, $g$ is continuously differentiable in $\mathbb R$, $a$ a constant real, how to calculate the derivative of $\int_a^{g(x)}f(x,t)dt$ with regard to $x$ (not by numerical methods)? Thanks.

Use Leibnitz's Rule for differentiation under the integral sign:

$\frac{\partial}{\partial x}\int\limits_a^{g(x)}f(x,t)\,dt=\int\limits_a^{g( x)}\frac{\partial f(x,t)}{\partial x}\,dx+f(x,g(x))\frac{\partial g(x)}{\partial x}-f(x,a)\frac{\partial a}{\partial x}$

Tonio

3. Thank you, erudite Tonio.

4. After reading the proof of this general Leibniz integral rule, I have to stress that the conditions supporting the special case of Leibniz integral rule(*) must be preserved to make possible the interchange of the lim and int process in producing the integral part of the result when $\Delta x\to 0$, the most important of which is the continuity of the partial derivative.
(*) $\frac{d}{dx}\int_a^b f(x,t)dt=\int_a^b \frac{\partial}{\partial x}f(x,t)dt$ where a,b are both constant.
ps: My question comes from the proof of Poincare's lemma in baby Rudin, the proof that $F\in\mathcal C'(E)$ is omitted as usual in Th10.38, I think the general Leibniz integral rule is necessary to prove it which is not mentioned at all in this book. I hope it is not the author's original intention to leave such a huge gap for me to fill. Thanks to Tonio I finally made it.