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


    \[ I(x)=\int_{a}^{b}f(t,x)\mathrm{d}t} \]


    \[ I'(x)= \int_{a}^{b}\frac{\partial \mathscr{F}}{\partial x}\mathrm{d}t} \]

Notice how this compares with the general case.


Here’s an example to stimulate curiosity in learning.

    \[ F(x)=\int_{1}^{2}\frac{cos(tx)}{t}\mathrm{d}t} \]

    \[ F^{\prime}(x)=\frac{d}{dx}\int_{1}^{2}\frac{cos(tx)}{t}\mathrm{d}t} \]

Now, by the special case of Leibniz Rule:

    \[ =\int_{1}^{2}\frac{\partial}{\partial x} \left[\frac{cos(tx)}{t} \right]\mathrm{d}t} \]

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

    \[ \follows \int_{1}^{2}\frac{-t\sin(tx)}{t}\mathrm{d}t} \]

Cancel, t, in numerator and denominator.

    \[ =\int_{1}^{2}-\sin(tx)\mathrm{d}t} \]

Note, we’re integrating wrt. t, therefore

    \[ =\left[\frac{cos(tx)}{x} \right]_{t=1}^{t=2} \]

    \[ = \frac{\cos(2x) - \cos(x)}{x} \]

As you can see, the special case is a simpler operation. The key thing to remember, again, is to insert the partial in the integral and then take the partial derivative of the integrand. From that point, it is usually just a matter of cleaning up the algebra.