Derivatives of $(z-a)^kf(z)$ at $a$ knowing that $f\in H(D(a,r)\setminus \{a\})$

Question

If $g(z)=(z-a)^nf(z)$ with $f\in H(D(a,r)\setminus \{a\})$.

Can we said that $g^{(k)}(a)=0$ for all $k\in \{0,1,...,n-1\}$?

Answer 1

$f(z)=\frac1{(z-a)^3}\in H(D(a,r)\setminus\{a\})$, yet the first derivative of $(z-a)^4f(z)$ is $1$ at $z=a$.