This paper introduces a forward differential deep learning algorithm (DLDBSDE) for solving high-dimensional nonlinear backward stochastic differential equations (BSDEs). The key innovation is transforming the BSDE into a differential deep learning problem using Malliavin calculus, which yields a system of BSDEs for the solution triple (Y, Z, Γ) representing the solution, its gradient, and Hessian matrix. The algorithm discretizes the system via Euler-Maruyama and employs three deep neural networks to approximate the unknown processes. The network parameters are optimized by minimizing a weighted sum of local loss functions that enforce the discretized dynamics of both Y and Z, incorporating explicit information about the gradient and Hessian. This approach contrasts with existing forward deep learning methods that rely solely on automatic differentiation for Z and Γ, often leading to less accurate gradient approximations. Numerical experiments on high-dimensional examples demonstrate that DLDBSDE achieves superior accuracy and computational efficiency, particularly for computing Γ, compared to the local deep BSDE scheme. The method is especially relevant for financial applications requiring precise delta and gamma hedging.