Partial derivative of integral: Leibniz rule?

Question

The Liebniz rule is as follows:

What I would like to know is how to apply the above formula for the case of the partial derivative:

$\displaystyle \ \ \frac{\partial}{\partial\alpha} \int_{a(\alpha)}^{b(\beta)} f(x,\alpha)dx$.

Thanks.

Answer 1

You compute a partial derivative with respect to $\alpha$ by holding $\beta$ fixed, and then just differentiating the resulting function of $\alpha$, which is a function of a single variable. And yes, the Liebniz rule tells you how to differentiate this function of $\alpha$.

For a given $\beta$, the derivative of the function \begin{align*} g(\alpha) &= \int_{a(\alpha)}^{b(\beta)} f(x,\alpha) \, dx \end{align*} is \begin{equation} \frac{dg(\alpha)}{d\alpha} = 0 - \frac{da(\alpha)}{d\alpha} f(a(\alpha),\alpha) + \int_{a(\alpha)}^{b(\beta)} \frac{\partial}{\partial \alpha} f(x,\alpha) \, dx. \end{equation}