We study the estimation of partial derivatives of nonparametric regression functions with many variables, with a view to conducting a significance test for the said derivatives. Our test is based on the moment generating function of the smoothed partial derivatives of an estimator of the regression function, where the estimator is a deep neural network. We demonstrate that in the context of modelling with neural networks, derivative estimation is in fact quite different from estimating the regression function itself, and hence the smoothing operation becomes important. To conduct an effective test with predictors of high or even diverging dimension, we assume that first, the observed high-dimensional predictors arise from a factor model and that second, only the lower-dimensional but latent factors and a subset of the marginals of the high-dimensional predictors drive the regression function. Moreover, we finely adjust the regression function estimator in order to achieve the desired asymptotic normality under the null hypothesis that the partial derivative in question is zero. We demonstrate the performance of our test in simulation studies.