  Non-perturbative features of the derivative expansion of the effective action
of a single D3-brane are obtained by considering scattering amplitudes of open
and closed strings. This motivates expressions for the coupling constant
dependence of world-volume interactions of the form $(\partial F)^4$ (where F
is the Born-Infeld field strength), $(\partial^2\phi)^4$ (where $\phi$ are the
normal coordinates of the D3-brane) and other interactions related by $\calN=4$
supersymmetry. These include terms that transform with non-trivial modular
weight under Montonen-Olive duality. The leading D-instanton contributions that
enter into these effective interactions are also shown to follow from an
explicit stringy construction of the moduli space action for the
D-instanton/D3-brane system in the presence of D3-brane open-string sources
(but in the absence of a background antisymmetric tensor potential). Extending
this action to include closed-string sources leads to a unified description of
non-perturbative terms in the effective action of the form (embedding
curvature)$^2$ together with open-string interactions that describe
contributions of the second fundamental form.
