Tobias Holck Colding continues MIT's 18.100B Real Analysis course with a lecture on differentiation built from rigorous first principles rather than computational shortcuts. He defines what it means for a function to be differentiable and proves that differentiability at a point implies continuity there. From that foundation he establishes the standard algebraic laws of differentiation, including Leibniz's product rule, the quotient rule, and the chain rule, each derived with full proofs rather than asserted. The lecture closes with two classical results: Rolle's lemma and the mean value theorem, both proved carefully and tied back to the earlier definitions. The style is chalkboard and formal argument throughout, aimed at students who already have the groundwork of limits and continuity from earlier in the course and are moving toward a proof based treatment of calculus.