Building on the observation that reverse-mode automatic differentiation (AD), a generalisation of backpropagation, can naturally be expressed as pullbacks of differential 1-forms, we design a higher-order, call-by-value functional programming language with a first-class reverse-mode AD operator. Using the category of convenient vector spaces and smooth maps, we show that the operational semantics precisely captures reverse-mode AD, even in a higher-order setting. We exhibit the connection with the differential λ-calculus via a translation that preserves reduction and interpretation.