Let $T_a$ be the first hitting time of level $a$ for standard Brownian motion. I am trying to show the density of $T_a$ is $f_a(t)= \frac{a}{\sqrt{2\pi t^3}}\text{exp}(-\frac{a^2}{2t})$
I know that $\mathbb{P}[T_a \leq t ] = 2\mathbb{P}[B_t \geq a]$ where $B_t \sim N(0,t)$ (standard brownian motion)
I have reduced this to: $F_a(t) = 2 \Phi(-\frac{a}{\sqrt{t}})$ where $\Phi$ denotes the cumulative density of the standard normal RV.
I did not have much luck with differentiation under the sign (aka Leibniz rule)
Is there a simple transformation that is more fruitful?
Your CDF is slightly wrong $P(T_a \le t) = 2 P(B_t \ge a) = 2 \Phi(-a/\sqrt{t})$.
Applying the Fundamental Theorem of Calculus yields $$ \frac{d}{dt} \left[2\int_{-\infty}^{-a/\sqrt{t}} \phi(u) \, du\right] = 2 \phi(-a/\sqrt{t}) \frac{d}{dt} (-a/\sqrt{t}) = at^{-3/2} \phi(-a/\sqrt{t}). $$