Master the theory of differentiation with rigorous definitions, comprehensive mean value theorems, powerful computational tools like L'Hospital's rule, and the fundamental Taylor expansion that connects calculus to infinite series.
Master the precise definition of derivatives, one-sided derivatives, relationship with continuity, and fundamental differentiation rules
Master the chain rule for composite functions, inverse function derivatives, implicit differentiation, parametric derivatives, and higher-order derivatives
Prove and apply Fermat's theorem, Rolle's theorem, Lagrange's mean value theorem, Cauchy's mean value theorem, and Darboux's theorem
Master L'Hospital's rule for indeterminate forms, Taylor's theorem with various remainder forms, and standard Taylor expansions
Apply derivatives to analyze monotonicity, extrema, convexity, inflection points, asymptotes, and function graphing
Developed calculus and fluxions notation
Developed calculus and modern notation
Mean Value Theorem, Taylor's theorem
L'Hospital's rule
Taylor series expansion
Rolle's theorem