This chapter defines the derivative as the limit of the difference quotient, representing the instantaneous rate of change and the slope of a tangent line. It establishes that while differentiability requires continuity, the presence of sharp corners, vertical tangents, or jump discontinuities can make a continuous function non-differentiable. The chapter details essential differentiation tools, including the Chain Rule, Implicit Differentiation, and formulas for trigonometric, exponential, and logarithmic functions. Practical applications focus on calculating one-sided derivatives, finding equations for tangent and normal lines, and determining the intersection angles between curves.