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The Problem Solver's Resource

Introduction \| Other Tips and Tricks \| Methods of Proof \| You are currently viewing page 9.

Derivatives

This 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

$\frac{df(x)}{dx}=\lim_{x\to x_0}\frac{f(x_0)-f(x)}{x_0-x}$ , where $f(x)$ is a function continuous in $L$ , and $x_0$ is an arbitrary constant such that $x_0\subset L$ .

Multiple derivatives are taken by evaluating the innermost first, and can be notated as follows: $\frac{d^2f(x)}{dx^2}$ .

The derivative of $f(x)$ can also be expressed as $f'(x)$ , or the $n$ th derivative of $f(x)$ can be expressed as $f^{(n)}(x)$ .

Basic Facts

$\frac{df(x)\pm g(x)}{dx}=f'(x)\pm g'(x)$
$\frac{df(x)\cdot g(x)}{dx}=f'(x)\cdot g(x)+ g'(x)\cdot f(x)$
$\frac{d\frac{f(x)}{g(x)}}{dx}=\frac{f'(x)g(x)-g'(x)f(x)}{g^2(x)}$
$\frac{d\sin x}{dx}=\cos x$
$\frac{d\cos x}{dx}=-\sin x$
$\frac{d\tan x}{dx}=\sec^2 x$
$\frac{d\csc x}{dx}=-\csc x\cot x$
$\frac{d\cot x}{dx}=-\csc^2 x$
$\frac{d\ln |x|}{dx}=\frac{1}{x}$

The Power Rule

$\frac{dx^n}{dx}=nx^{n-1}$

Rolle's Theorem

If $f(x)$ is differentiable in the open interval $(a,b)$ , continuous in the closed interval $[a,b]$ , and if $f(a)=f(b)$ , then there is a point $c$ between $a$ and $b$ such that $f'(c)=0$

Extension: Mean Value Theorem

If $f(x)$ is differentiable in the open interval $(a,b)$ and continuous in the closed interval $[a,b]$ , then there is a point $c$ between $a$ and $b$ such that $f(b)-f(a)=f'(c)\cdot(b-a)$ .

L'Hopital's Rule

$\lim \frac{f(x)}{g(x)}=\lim \frac{f'(x)}{g'(x)}$

Note that this inplies that $\lim \frac{f(x)}{g(x)}=\lim \frac{f^{(n)}(x)}{g^{(n)}(x)}$ for any $n$ .

Taylor's Formula

Let $a$ be a point in the domain of the function $f(x)$ , and suppose that $f^{(n+1)}(x)$ (that is, the $n+1$ th derivative of $f(x)$ ) exists in the neighborhood of $a$ (where $n$ is a nonnegative integer). For each $x$ in the neighborhood,

$f(x)=f(a)+f'(a)(x-a)+\frac{f''(a)}{2!}(x-a)^2+...+\frac{f^{(n)}(a)}{n!}(x-a)^n+\frac{f^{(n+1)}(a)}{n!}(x-a)^{n+1}$

where $c$ is in between $x$ and $a$ .

Chain Rule

If $h(x) = f(g(x))$ , then $h'(x)=f'(g(x))\cdot g'(x)$

Applications

The slope of $f(x)$ at any given point is the derivative of $f(x)$ . (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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@@ Line 25: / Line 25: @@
 *<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>
 ====The Power Rule====
 *<math>\frac{dx^n}{dx}=nx^{n-1}</math>

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