= Solution
A <kernel for density estimation> is an integrable function $K$ with $\int K=1$. The <kernel density estimator> is
$$
\widehat f_{n,h,K}(x)
=\frac1{nh}\sum_{i=1}^nK\left(\frac{x-X_i}{h}\right).
$$
It has order $\ell$ when $\int u^jK(u)\,du=0$ for $1\leq j<\ell$ and $\int|u|^\ell|K(u)|\,du<\infty$.
Differentiation gives the <derivative kernel density estimator>
$$
\widehat f_n'(x)=\frac1{nh^2}\sum_{i=1}^nK'\left(\frac{x-X_i}{h}\right).
$$
Dropping the negative square of its mean from the variance and using Tonelli and the substitution $u=(x-X_1)/h$,
$$
\begin{aligned}
\int_{\mathbb R}\operatorname{Var}\widehat f_n'(x)\,dx
&\leq\frac1{nh^4}\int_{\mathbb R}
\mathbb E K'\left(\frac{x-X_1}{h}\right)^2dx\\
&=\frac1{nh^3}\int_{-1}^1K'(u)^2du.
\end{aligned}
$$
Thus $\alpha=3$ and $C_1(K)=\lVert K'\rVert_2^2$.
For $r=\lfloor\beta\rfloor$, the <Nikolsky class> $\mathcal N(\beta,L)$ consists of functions $g$ with square-integrable derivatives through order $r$ and
$$
\lVert g^{(r)}(\mathord\cdot+t)-g^{(r)}\rVert_2
\leq L|t|^{\beta-r},
$$
using the equivalent finite-difference definition when $\beta$ is an integer. Integration by parts shows
$$
\mathbb E\widehat f_n'=K_h*f'.
$$
Taylor expansion in $L^2$, cancellation of the moments through order $\ell-1$ for $\ell=\lceil\beta\rceil$, and the generalized Minkowski inequality give
$$
\lVert K_h*f'-f'\rVert_2
\leq
\frac{Lh^\beta}{r!}\int|u|^\beta|K(u)|\,du.
$$
Consequently
$$
\int_{\mathbb R}\operatorname{Bias}\{\widehat f_n'(x)\}^2dx
\leq C_2(\beta,L,K)h^{2\beta},
$$
where one admissible choice is
$$
C_2(\beta,L,K)
=\frac{L^2}{(r!)^2}
\left(\int|u|^\beta|K(u)|\,du\right)^2.
$$
Hence $\gamma=2\beta$.
The mean integrated squared error is
$$
\operatorname{MISE}(\widehat f_n')
=\mathbb E\int_{\mathbb R}\{\widehat f_n'(x)-f'(x)\}^2dx,
$$
the sum of integrated variance and squared bias. The preceding bounds give
$$
\operatorname{MISE}(\widehat f_n')
\leq\frac{C_1(K)}{nh^3}+C_2(\beta,L,K)h^{2\beta}.
$$
Taking $h\asymp n^{-1/(2\beta+3)}$ proves the <MISE rate for derivative kernel density estimation>
$$
\inf_{h>0}\sup_{f:f'\in\mathcal N(\beta,L)}
\operatorname{MISE}(\widehat f_n')
\leq C_3(\beta,L,K)n^{-2\beta/(2\beta+3)}.
$$
Thus $\delta=2\beta/(2\beta+3)$. Estimating a density of the same smoothness has variance order $(nh)^{-1}$ and rate $n^{-2\beta/(2\beta+1)}$; estimating its derivative is harder because differentiation amplifies high-frequency noise, changing $h^{-1}$ to $h^{-3}$.
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