CAS MA 123 Lecture Notes - Lecture 9: Quotient Rule, Product Rule

• School: Boston University
• Department: Mathematics & Statistics
• Course: CAS MA 123
• Professor: John Bergdall

Document Summary

Let e=2. 719 (approximately), then f(x) =ex satisfies f"(x) = f(x) i. e. d dx (ex)= ex: so the derivative is the original function. Consider the graph of x ax as the box a is varying: Think about the tangent line to y=ax @ point (0,1: slope is given by the limit lim h 0 ah 1 h. = l(a: when a=1, the slope=0 when a approaches infinity the slope increases to infinity, the slope to y=ax @ (0,1) is continuous on variable a and increasing strictly. Conclusion: there is a unique h, we call it e, so that the slope of the tangent= lim h 0 eh 1 h. As the exponential property ex+h = ex * eh, then d dx (ex) = lim h 0 eh +x ex h. *where ex is a constant with respect to h* *this also means ex is very special*