Calculus Anyone? — OZONE Asylum, home of the Mad Scientists

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Your Text: Insert Slimies » Insert UBB Code » Close Last Tag \| All Tags UBB Help	Hyperbole's derivation is an approach from first principles - this is the actual definition of the limit, and where all differentiation comes from. E.g. for the general polynomial x^n: d/dx(x^n)=lim(h->0)( (x+h)^n - x^n )/h (with appropriate assumptions about convergence) = lim(h->0)(x^n + hnx^n-1 + ... + h^n - x^n)/h = lim(h->0)( nhx^n-1 + O(h^2) )/h = nx^n-1. The product rule is a similar idea: lim(h->0) (f(x+h)g(x+h))-(f(x)g(x))/h = lim(h->0) (f(x+h)g(x+h)-f(x+h)g(x)+f(x+h)g(x)-f(x)g(x))/h (since -f(x+h)g(x)+f(x+h)g(x)=0) = lim(h->0) f(x+h)g'(x) + g(x)f'(x) = f(x)g'(x)+g(x)f'(x) (again, with sufficient conditions on existence of differentials) However, the quotient rule shows why the conditions are necessary: it's very tempting to simply sub in g(x)=k(x)^-1 for some k(x) and use the chain rule to produce the answer. This works in the majority of cases, but requires extra rigour to handle the exceptional cases - in practice, a slightly more fiddly proof is used for the quotient rule. (analysis is fun ^_^) [url=http://www.wrayal.org][img]http://www.wrayal.org/JS/asylum.JPG[/img][/url] Loading...
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