Find all values of a and b that make the following function differentiable for all values of x

Problem
Find all values of $a$ and $b$ that make the following function differentiable for all values of $x$: $$f(x) = \begin{cases} \arctan(ax+b), x<0\\ \pi/4e^{\sin(bx)}, x \geq 0\\ \end{cases}$$

I thought I had this question figured out but it started to get more complicated than I think it should be. Any help would be greatly appreciated, Thank you.

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$\arctan(b) = \frac{\pi}{4}$ implies that $b = 1$.
$\frac{a}{1+(ax+b)^2} = \frac{a}{1+1}$ set equal to $\frac{\pi}{4}b =\frac{\pi}{4}$. So $a = \frac{\pi}{2}$