I am trying to solve the following $m$th-derivative of standard normal cdf, $$\frac{\text{d}^m}{\text{d}a^m}\Phi \left(\frac{a+\mu u}{\sqrt{u}}\right),$$ where $m> 0$ is an integer , $\mu>0$, and $u>0$ are constants. $a>0$.
After trying to do this with Maple, found it hard to reach a closed form of solution.The length of derivative increases as $m$ increases.
This problem comes from $$\begin{split} &\frac{\text{d}^m}{\text{d}a^m}\left(\int_{0}^{u}\frac{a}{\sqrt{2\pi x^3}} e^{-\frac{(a+\mu x)^2}{2x}}dx \right)\\ = & \frac{\text{d}^m}{\text{d}a^m}\left( \mathbb{P}(\tau_{a}<u)\right)\\ = & \frac{\text{d}^m}{\text{d}a^m}\left(\mathbb{P}(\max_{0\leq s\leq u} (W_{t}-\mu t)\geq a)\right)\\ = & \frac{\text{d}^m}{\text{d}a^m} \left( 1- \Phi (\frac{a+\mu u}{\sqrt{u}})+e^{-2\mu a}\bar{\Phi}(\frac{a-\mu u}{\sqrt{u}}) \right) \end{split},$$ where $\tau _a$ is the first hitting time when the drifted Brownian motion $W_t-\mu t$ touches the barrier $a$.
Any help or idea for this problem would be the most grateful.
Edit: The hint provided by Did in the comment is very helpful and does work, since $$\frac{\text{d}^m}{\text{d}x^m}\Phi \left(x\right)=(-1)^{m-1}H_{m-1}(x)\phi(x).$$
Just got another question about the following multi-derivative $$\frac{\text{d}^m}{\text{d}a^m} \left(e^{-2\mu a}\Phi \left(\frac{a-\mu u}{\sqrt{u}}\right)\right),$$ I tried to find $$\frac{\text{d}^m}{\text{d}x^m} \left(e^{bx}\Phi \left(x\right)\right)$$ firstly, but it seems be very difficult to get a recursive answer like the first multi-derivative.
Any help or idea would be the most grateful.