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The Problem Solver's Resource

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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

$\displaystyle\frac{df(x)}{dx}=\lim_{x->x_0}\frac{f(x_0)-f(x)}{x_0-x}$ , 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 $f^{(n)}(x)$ .

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 $a$ . Back to page 8 | Continue to page 10

@@ Line 6: / Line 6: @@
 | style="background:lime; border:1px solid black;height:200px;padding:10px;" | {{User:Temperal/testtemplate|page 9}}
 ==<span style="font-size:20px; color: blue;">Derivatives</span>==
-<!-- will fill in later! -->
+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>\displaystyle\frac{df(x)}{dx}=\lim_{x->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>.
+===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>.
 [[User:Temperal/The Problem Solver's Resource8|Back to page 8]] | [[User:Temperal/The Problem Solver's Resource10|Continue to page 10]]
 |}<br /><br />

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