This chapter examines gradient-based optimization methods, essential tools in modern machine learning and artificial intelligence. We extend previous optimization approaches to continuous spaces, showing how derivatives guide the search process toward optimal solutions. The material progresses from fundamental gradient descent to sophisticated adaptive techniques, with emphasis on neural network training applications. Key concepts include learning rate dynamics, momentum-based acceleration, and specialized algorithms like Adam and RMSprop. Each method is presented with mathematical foundations, implementation details, and convergence analysis. Practical Python implementations demonstrate algorithm behavior on both classical optimization problems and contemporary machine learning tasks. The chapter concludes with emerging research directions and advanced techniques for large-scale systems. Through this structured approach, readers gain both theoretical understanding and practical implementation skills for applying gradient-based methods to complex optimization challenges.