I'm attempting Exercise 5.1.15 of Witherspoon's Hochschild cohomology for Algebras. For an algebra $(A,\cdot)$, a deformation of it (over $\mathbb{C}[t]$) will be a product $\star$ on $A\otimes \mathbb{C}[t]$ such that $r\star s=r\cdot s+\mu_1(r\otimes s)t+\mu_2(r\otimes s)t^2+\dots$ for some $\mathbb{C}$-linear functions $\mu_i:A\otimes A\rightarrow A$. Associativity of the deformation implies $\mu_1$ is a Hochschild $2$-cocycle of $A$. Now $\mathbb{C}[x,y]$ has a deformation which is (on quotienting by the ideal generated by $t-1$) the Weyl algebra $A_1:=\mathbb{C}\langle x,y\rangle / \langle xy-yx-1\rangle$. We're asked to find $\mu_1$ and $\mu_2$ of this deformation.
I've been considering the product $x^ay^b\star x^cy^d$ in the deformed algebra with relation $y\star x=x\star y +t$. On experimenting with various choices of $a,b,c,d$, I think I have part of a general formula: $$x^ay^b\star x^cy^d=x^{a+c}y^{b+d}+bcx^{a+c-1}y^{b+d-1}t+?x^{a+c-2}y^{b+d-2}t^2+\dots$$ I'm not sure what the coefficient marked by ? is yet. So $\mu_1(x^ay^b\otimes x^cy^d)=bcx^{a+c-1}y^{b+d-1}$ and $\mu_i\sim \mu_{i-1}/xy$. I would appreciate any advice on how this can be proven, or at least confirm if my guess is on the right lines, thanks!