Leibniz’s Formula
Tuesday, May 1st, 2007 -- By ETSuppose that ![\displaystyle f(t,x), a(t)[/tex], and](http://blog.mikezhang.com/wp-content/cache/tex_f2119ff6bff73f42bec17c2ca1db2eed.png)
\displaystyle b(t)[/tex] are differentiable functions, and that
![\displaystyle F(t) = \int_{a(t)}^{b(t)}{f(t,x)}dx[/tex]<br />
Then the derivative of](http://blog.mikezhang.com/wp-content/cache/tex_007034d21db6df08442b53728b59bbd4.png)
\displaystyle F[/tex] is given by
$$\displaystyle F^{’}(t)=f\left(t,b(t)\right)b^{’}(t)-f\left(t,a(t)\right)a^{’}(t)+\int_{a(t)}^{b(t)}{\displaystyle \frac{\partial f(t,x)}{\partial t}}dx[/tex]
