(a) Let $A$ be a commutative algebra over a field $k$, and $p: A \rightarrow k$ a $k$-linear homomorphism. Define $\operatorname{Der}(A, p)$, the derivations of $A$ centered in $p$, and define the tangent space $T_{p} A$ in terms of this.

Show directly from your definition that if $f \in A$ is not a zero divisor and $p(f) \neq 0$, then the natural map $T_{p} A\left[\frac{1}{f}\right] \rightarrow T_{p} A$ is an isomorphism.

(b) Suppose $k$ is an algebraically closed field and $\lambda_{i} \in k$ for $1 \leqslant i \leqslant r$. Let

$$X=\left\{(x, y) \in \mathbb{A}^{2} \mid x \neq 0, y \neq 0, y^{2}=\left(x-\lambda_{1}\right) \cdots\left(x-\lambda_{r}\right)\right\}$$

Find a surjective map $X \rightarrow \mathbb{A}^{1}$. Justify your answer.