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==<span style="font-size:20px; color: blue;">Derivatives</span>==
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==<span style="font-size:20px; color: blue;">Complex Numbers</span>==
This page will cover derivatives and their applications, as well as some advanced limits. The Fundamental Theorem of Calculus is covered on the [[User:Temperal/The Problem Solver's Resource10|integral page]].
 
 
 
===Definition===
 
 
 
*<math>\frac{df(x)}{dx}=\lim_{x\to x_0}\frac{f(x_0)-f(x)}{x_0-x}</math>, where <math>f(x)</math> is a function continuous in <math>L</math>, and <math>x_0</math> is an arbitrary constant such that <math>x_0\subset L</math>.
 
 
 
*Multiple derivatives are taken by evaluating the innermost first, and can be notated as follows: <math>\frac{d^2f(x)}{dx^2}</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>
 
*<math>\frac{d\sin x}{dx}=\cos x</math>
 
*<math>\frac{d\cos x}{dx}=-\sin x</math>
 
*<math>\frac{d\tan x}{dx}=\sec^2 x</math>
 
*<math>\frac{d\csc x}{dx}=-\csc x\cot x</math>
 
*<math>\frac{d\cot x}{dx}=-\csc^2 x</math>
 
*<math>\frac{d\ln |x|}{dx}=\frac{1}{x}</math>
 
*<math>\frac{dc}{dx}=0</math> for a constant <math>c</math>.
 
====The Power Rule====
 
*<math>\frac{dx^n}{dx}=nx^{n-1}</math>
 
 
 
===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>
 
====Extension: Mean Value Theorem====
 
If <math>f(x)</math> is differentiable in the open interval <math>(a,b)</math> and continuous in the closed interval <math>[a,b]</math>, then there is a point <math>c</math> between <math>a</math> and <math>b</math> such that <math>f(b)-f(a)=f'(c)\cdot(b-a)</math>.
 
===L'Hopital's Rule===
 
<math>\lim \frac{f(x)}{g(x)}=\lim \frac{f'(x)}{g'(x)}</math>
 
 
 
Note that this inplies that <math>\lim \frac{f(x)}{g(x)}=\lim \frac{f^{(n)}(x)}{g^{(n)}(x)}</math> for any <math>n</math>.
 
===Taylor's Formula===
 
Let <math>a</math> be a point in the domain of the function <math>f(x)</math>, and suppose that <math>f^{(n+1)}(x)</math> (that is, the <math>n+1</math>th derivative of <math>f(x)</math>) exists in the neighborhood of <math>a</math> (where <math>n</math> is a nonnegative integer). For each <math>x</math> in the neighborhood,
 
 
 
<cmath>f(x)=f(a)+f'(a)(x-a)+\frac{f''(a)}{2!}(x-a)^2+...+\frac{f^{(n)}(a)}{n!}(x-a)^n+\frac{f^{(n+1)}(a)}{n!}(x-a)^{n+1}</cmath>
 
 
 
where <math>c</math> is in between <math>x</math> and <math>a</math>.
 
===Chain Rule===
 
If <math>h(x) = f(g(x))</math>, then <math>h'(x)=f'(g(x))\cdot g'(x)</math>
 
===Applications===
 
*The slope of <math>f(x)</math> at any given point is the derivative of <math>f(x)</math>. (The obvious one.)
 
*Acceleration is the derivative of velocity in relation to time; velocity is the derivative of position in relation to time.
 
*The derivative of work (in Joules) in relation to time is power (in watts).
 
  
 
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Revision as of 22:19, 10 January 2009

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