Derivation of telegrapher's equations
A better way to look at it is through the Fourier transform, which is what using phasors does indirectly. Hence

$$\int_{-\infty}^\infty \frac{\partial v}{\partial z} e^{-j\omega t} \ \mathrm{d}t = \frac{\partial}{\partial z} \int_{-\infty}^{\infty} v(z, t) e^{-j\omega t} \ \mathrm{d} t = \frac{\partial V}{\partial z} (z, \omega),$$

where \$V(z, \omega)\$ is the Fourier transform of \$v(z, t)\$. Looking at a single frequency \$\omega_0\$ yields

$$\frac{\partial V}{\partial z}(z, \omega_0) = \frac{\mathrm{d} V_{\omega_0}}{\mathrm{d} z} (z),$$

where \$V_{\omega_0} = V(z, \omega_0)\$ is a single variable function that depends only on \$z\$.