$$\begin{aligned}& \biggl(\frac{\partial f}{\partial z_{1}} \biggr)^{\land}(\lambda,\sigma)g(u) \\& \quad =\int_{\mathbb{R}^{n}}\int_{K}{ \frac{\partial f}{\partial z_{1}}(z_{1},z_{2},\ldots,z_{n},k) \pi_{\lambda,\sigma}(z_{1},z_{2},\ldots ,z_{n},k)^{\ast}g(u)} \,dz_{1} \,dz_{2} \cdots \,dz_{n} \,dk \\& \quad =\int_{\mathbb{R}^{n}}\int_{K}\lim _{h\rightarrow0}{ \biggl[\frac {f(z_{1}+h,z_{2},\ldots,z_{n},k)-f(z_{1},z_{2},\ldots,z_{n},k)}{h} \biggr]} \\& \qquad {}\times\pi _{\lambda,\sigma}(z_{1},z_{2},\ldots,z_{n},k)^{\ast}g(u)\,dz_{1} \,dz_{2} \cdots \,dz_{n} \,dk \\& \quad =\lim_{h\rightarrow0}\frac{1}{h}\biggl[\int _{\mathbb{R}^{n}}\int_{K}{f(z_{1}+h,z_{2}, \ldots,z_{n},k) \pi_{\lambda,\sigma}(z_{1},z_{2}, \ldots ,z_{n},k)^{\ast}g(u)} \,dz_{1} \,dz_{2} \cdots \,dz_{n} \,dk \\& \qquad {}-\int_{\mathbb{R}^{n}}\int_{K}{f(z_{1},z_{2}, \ldots ,z_{n},k) \pi _{\lambda,\sigma}(z_{1},z_{2}, \ldots,z_{n},k)^{\ast}g(u)} \,dz_{1} \,dz_{2} \cdots \,dz_{n} \,dk\biggr] \\& \quad =\lim_{h\rightarrow0}\frac{1}{h}\biggl[\int _{\mathbb{R}^{n}}\int_{K}f(z_{1},z_{2}, \ldots,z_{n},k) e^{-i \lambda h u_{11}}\pi_{\lambda ,\sigma}(z_{1},z_{2}, \ldots,z_{n},k)^{\ast}\\& \qquad {}\times g(u) \,dz_{1} \,dz_{2} \cdots \,dz_{n} \,dk \\& \qquad {}-\int_{\mathbb{R}^{n}}\int_{K}{f(z_{1},z_{2}, \ldots ,z_{n},k) \pi _{\lambda,\sigma}(z_{1},z_{2}, \ldots,z_{n},k)^{\ast}g(u)} \,dz_{1} \,dz_{2} \cdots \,dz_{n} \,dk\biggr] \\& \quad =\lim_{h\rightarrow0} \biggl[\frac{e^{-i \lambda h u_{11}}-1}{h} \biggr]\int _{\mathbb{R}^{n}}\int_{K}f(z_{1},z_{2}, \ldots ,z_{n},k) \pi_{\lambda,\sigma}(z_{1},z_{2}, \ldots,z_{n},k)^{\ast}\\& \qquad {}\times g(u) \,dz_{1} \,dz_{2} \cdots \,dz_{n} \,dk \\& \quad =i\lambda u_{11}\int_{\mathbb{R}^{n}}\int _{K}{f(z_{1},z_{2},\ldots ,z_{n},k) \pi _{\lambda,\sigma}(z_{1},z_{2}, \ldots,z_{n},k)^{\ast}g(u)} \,dz_{1} \,dz_{2} \cdots \,dz_{n} \,dk \\& \quad =i\lambda u_{11} \hat{f}(\lambda,\sigma)g(u). \end{aligned}$$