text: Using the definition above for  and , the first and second derivatives can be expressed in terms of the Faddeeva function as
  and
  respectively.
  Often, one or multiple Voigt profiles and/or their respective derivatives need to be fitted to a measured signal by means of non-linear least squares, e.g., in spectroscopy. Then, further partial derivatives can be utilised to accelerate computations. Instead of approximating the Jacobian matrix with respect to the parameters , , and  with the aid of finite differences, the corresponding analytical expressions can be applied. With  and , these are given by:
  for the original voigt profile ;
  for the first order partial derivative ; and
  for the second order partial derivative . Since  and  play a relatively similar role in the calculation of , their respective partial derivatives also look quite similar in terms of their structure, although they result in totally different derivative profiles. Indeed, the partial derivatives with respect to  and  show more similarity since both are width parameters. All these derivatives involve only simple operations (multiplications and additions) because the computationally expensive  and  are readily obtained when computing . Such a reuse of previous calculations allows for a derivation at minimum costs. This is not the case for finite difference gradient approximation as it requires the evaluation of  for each gradient respectively.