I recently wrote a short piece on the general case of the Leibniz Rule. In that article I alluded to how there is also a special case of the rule, in which the limits of integration are constants as opposed to differentiable functions.

In short, the special case of the Leibniz Rule is a much simpler operation. The key idea remains that one must perform partial differentiation within the the integral, but the rest of the general case is cut out. This means all one has to do is perform any remaining basic integration techniques and then clean up the evaluation with some algebra.

In general, the special case of Leibniz Rule states

if

*** QuickLaTeX cannot compile formula:
$I(x)=\int_{a}^{b}f(t,x)\mathrm{d}t}$

*** Error message:
Extra }, or forgotten $. leading text: I(x)=\int_{a}^{b}f(t,x)\mathrm{d}t}  then *** QuickLaTeX cannot compile formula: $I'(x)= \int_{a}^{b}\frac{\partial \mathscr{F}}{\partial x}\mathrm{d}t}$ *** Error message: Undefined control sequence \mathscr. leading text: ...^{b}\frac{\partial \mathscr{F}}{\partial x}  Notice how this compares with the general case. *** Here’s an example to stimulate curiosity in learning. *** QuickLaTeX cannot compile formula: $F(x)=\int_{1}^{2}\frac{cos(tx)}{t}\mathrm{d}t}$ *** Error message: Extra }, or forgotten$.


*** QuickLaTeX cannot compile formula:
$F^{\prime}(x)=\frac{d}{dx}\int_{1}^{2}\frac{cos(tx)}{t}\mathrm{d}t}$

*** Error message:
Extra }, or forgotten $. leading text: ...x}\int_{1}^{2}\frac{cos(tx)}{t}\mathrm{d}t}  Now, by the special case of Leibniz Rule: *** QuickLaTeX cannot compile formula: $=\int_{1}^{2}\frac{\partial}{\partial x} \left[\frac{cos(tx)}{t} \right]\mathrm{d}t}$ *** Error message: Extra }, or forgotten$.



At this point, take the partial derivative. Working this out,

*** QuickLaTeX cannot compile formula:
$\follows \int_{1}^{2}\frac{-t\sin(tx)}{t}\mathrm{d}t}$

*** Error message:
Undefined control sequence \follows.



Cancel, t, in numerator and denominator.

*** QuickLaTeX cannot compile formula:
$=\int_{1}^{2}-\sin(tx)\mathrm{d}t}$

*** Error message:
Extra }, or forgotten \$.