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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

$\frac{df(x)\pm g(x)}{dx}=f'(x)\pm g'(x)$
$\frac{df(x)\cdot g(x)}{dx}=f'(x)\cdot g(x)+ g'(x)\cdot f(x)$
$\frac{d\frac{f(x)}{g(x)}}{dx}=\frac{f'(x)g(x)-g'(x)f(x)}{g^2(x)}$
$\frac{d\sin x}{dx}=\cos x$
$\frac{d\cos x}{dx}=-\sin x$
$\frac{d\tan x}{dx}=\sec^2 x$
$\frac{d\csc x}{dx}=-\csc x\cot x$
$\frac{d\cot x}{dx}=-\csc^2 x$
$\frac{d\ln |x|}{dx}=\frac{1}{x}$
$\frac{dc}{dx}=0$ for a constant $c$ .

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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