When time derivative and space derivatives can be interchanged?
Answers
In calculus, Leibniz's rule for differentiation under the integral sign, named after Gottfried Leibniz, states that for an integral of the form
{\displaystyle \int _{a(x)}^{b(x)}f(x,t)\,dt,} {\displaystyle \int _{a(x)}^{b(x)}f(x,t)\,dt,}
where {\displaystyle -\infty <a(x),b(x)<\infty } {\displaystyle -\infty <a(x),b(x)<\infty }, the derivative of this integral is expressible as
{\displaystyle {\frac {d}{dx}}\left(\int _{a(x)}^{b(x)}f(x,t)\,dt\right)=f{\big (}x,b(x){\big )}\cdot {\frac {d}{dx}}b(x)-f{\big (}x,a(x){\big )}\cdot {\frac {d}{dx}}a(x)+\int _{a(x)}^{b(x)}{\frac {\partial }{\partial x}}f(x,t)\,dt,} {\displaystyle {\frac {d}{dx}}\left(\int _{a(x)}^{b(x)}f(x,t)\,dt\right)=f{\big (}x,b(x){\big )}\cdot {\frac {d}{dx}}b(x)-f{\big (}x,a(x){\big )}\cdot {\frac {d}{dx}}a(x)+\int _{a(x)}^{b(x)}{\frac {\partial }{\partial x}}f(x,t)\,dt,}
where the partial derivative indicates that inside the integral, only the variation of f(x, t) with x is considered in taking the derivative.[1] Notice that if {\displaystyle a(x)} a(x) and {\displaystyle b(x)} b(x) are constants rather than functions of {\displaystyle x} x, we have a special case of Leibniz's rule:
{\displaystyle {\frac {d}{dx}}\left(\int _{a}^{b}f(x,t)\,dt\right)=\int _{a}^{b}{\frac {\partial }{\partial x}}f(x,t)\,dt.} {\displaystyle {\frac {d}{dx}}\left(\int _{a}^{b}f(x,t)\,dt\right)=\int _{a}^{b}{\frac {\partial }{\partial x}}f(x,t)\,dt.}
Thus under certain conditions, one may interchange the integral and partial differential operators. This important result is particularly useful in the differentiation of integral transforms. An example of such is the moment generating function in probability theory, a variation of the Laplace transform, which can be differentiated to generate the moments of a random variable. Whether Leibniz's integral rule applies is essentially a question about the interchange of limits.