Difference between revisions of "User:Temperal/The Problem Solver's Resource9"
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*The derivative of <math>f(x)</math> can also be expressed as <math>f'(x)</math>, or the <math>n</math>th derivative of <math>f(x)</math> can be expressed as <math>f^{(n)}(x)</math>. | *The derivative of <math>f(x)</math> can also be expressed as <math>f'(x)</math>, or the <math>n</math>th derivative of <math>f(x)</math> can be expressed as <math>f^{(n)}(x)</math>. | ||
+ | ===Basic Facts=== | ||
+ | *<math>\frac{df(x)\pm g(x)}{dx}=f'(x)\pm g'(x)</math> | ||
+ | *<math>\frac{df(x)\cdot g(x)}{dx}=f'(x)\cdot g(x)+ g'(x)\cdot f(x)</math> | ||
+ | *<math>\frac{d\frac{f(x)}{g(x)}}{dx}=\frac{f'(x)g(x)-g'(x)f(x)}{g^2(x)}</math> | ||
===Rolle's Theorem=== | ===Rolle's Theorem=== | ||
If <math>f(x)</math> is differentiable in the open interval <math>(a,b)</math>, continuous in the closed interval <math>[a,b]</math>, and if <math>f(a)=f(b)</math>, then there is a point <math>c</math> between <math>a</math> and <math>b</math> such that <math>f'(c)=0</math> | If <math>f(x)</math> is differentiable in the open interval <math>(a,b)</math>, continuous in the closed interval <math>[a,b]</math>, and if <math>f(a)=f(b)</math>, then there is a point <math>c</math> between <math>a</math> and <math>b</math> such that <math>f'(c)=0</math> |
Revision as of 14:00, 7 October 2007
DerivativesThis page will cover derivatives and their applications, as well as some advanced limits. The Fundamental Theorem of Calculus is covered on the integral page. Definition
Basic FactsRolle's TheoremIf Extension: Mean Value TheoremIf L'Hopital's Rule
Note that this inplies that Taylor's FormulaLet
where Applications
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