On page 27 of Fourier Analysis and Hausdorff Dimension, it seems like Mattila makes the claim that when $\Psi(x)=e^{-\pi x^2}$, then $\Psi = \widehat{\Psi}$. This is said to be true because both expressions satisfy the differential equation $f'(x)=-2\pi x f(x)$ with initial condition $f(0)=1$. I see why $\Psi$ satisfies this equation, but why is it that
$$\frac{d}{dt} \int_\mathbb{R} e^{-\pi x^2-2 \pi i x t} dx = -2 \pi x \int_\mathbb{R} e^{-\pi x^2-2 \pi i x t} dx?$$
Is this an issue with taking the derivative of a complex function?