Let $f:\mathbb{C} \to \mathbb{C}$ be an entire function of order $\rho$, that is, $\rho$ is the least positive real so that there are constants $x,y$ with $|f(z)| \leq xe^{y|z|^\rho}$ for every $z \in \mathbb{C}$. I would like to show that the derivative $f'(z)$ is of finite order $\rho' \leq \rho$.
My attempt: Since $f$ is entire, it is analytic, so we can write $$f(z) = \sum_{n \geq 0}a_n(z-z_0)^n$$
$$f'(z) = \sum_{n \geq 1}na_n(z-z_0)^{n-1} = \sum_{n\geq 0}(n+1)a_{n+1}(z-z_0)^n$$ Then using Cauchy estimates for the coefficients of $f'$, $$(n+1)|a_{n+1}| \leq \int_0^{2\pi}|f'(z_0+e^{i\theta})|\frac{d\theta}{2\pi}$$ My idea was to integrate the RHS to get a function involving only $f$, and use $\rho$ to put a bound on the coefficients of $f'$. I am having trouble proceeding. How might I approach this problem?