| 147 |
derivative of a function
|
\[
f^\prime\left(x\right)=\lim\limits_{h\rightarrow0}\frac{f\left(x+h\right)-f\left(x\right)}{h}
\]
|
| 149 |
Notations for the derivative
|
\[
y',\frac{dy}{dx},\frac{df}{dx},\frac{d}{dx}f\left(x\right)
\]
|
| 149 |
Notation for calculating a derivative
|
\[
f'\left(a\right)=\left.\frac{dy}{dx}
\right|_{x=a}=\lim\limits_{h\rightarrow0}
\frac{f\left(a+h\right)-f\left(a\right)}{h}
\]
|
| 149 |
derivative of a constant function
for $k$ constant
|
\[
\frac{d}{dx}\left(k\right)=0
\]
|
| 150 |
power rule for positive integers
|
\[
\frac{d}{dx}x^n=nx^{n-1}
\]
|
| 150 |
constant multiple rule
|
\[
\frac{d}{dx}\left(cu\right)=c\frac{du}{dx}
\]
|
| 151 |
derivative sum/difference rule
|
\[
\frac{d}{dx}\left(u\pm v\right)=\frac{du}{dx}\pm\frac{dv}{dx}
\]
|
| 152 |
derivative of a finite sum (use this rule to
evaluate the derivative of a polynomial)
|
\[
\frac{d}{dx}\sum_{i=1}^{n}u_i=\sum_{i=1}^{n}\frac{du_i}{dx}
\]
|
| 152 |
finding the horizontal tangents of a function.
Solve.
|
\[
\frac{dy}{dx}=0.
\]
|
| 153 |
right-hand differentiable
|
\[
\lim\limits_{h\rightarrow0^+}\frac{f\left(x+h\right)-f\left(x\right)}{h}
\]
|
| 153 |
left-hand differentiable
|
\[
\lim\limits_{h\rightarrow0^-}\frac{f\left(x+h\right)-f\left(x\right)}{h}
\]
|
| 156 |
symbols for derivatives
("$y$
super $n$" for
$y^{\left(n\right)}$)
|
\[
y',
y'',
\frac{d^2y}{dx^2},
y^{\left(n\right)},
\frac{d^ny}{dx^n}
\]
|
| 161 |
position function
|
\[
s=f\left(t\right)
\]
|
| 161 |
displacement
|
\[
\Delta s=f\left(t+\Delta t\right)-f\left(t\right)
\]
|
| 161 |
average velocity
|
\[
v_{av}=\frac{\Delta s}{\Delta t}
=\frac{f\left(t+\Delta t\right)-f\left(t\right)}
{\Delta t}
\]
|
| 161 |
instantaneous velocity
|
\[
\begin{split}
v\left(t\right)
=\frac{ds}{dt}
=\lim\limits_{\Delta t\rightarrow0}v_{av}
\end{split}
\]
|
| 162 |
speed
|
\[
\abs{v\left(t\right)}=\abslr{\frac{ds}{dt}}
\]
|
| 163 |
acceleration
|
\[
a\left(t\right)=\frac{dv}{dt}=\frac{d^2s}{dt^2}
\]
|
| 163 |
jerk
|
\[
j\left(t\right)=\frac{da}{dt}=\frac{d^3s}{dt^3}
\]
|
| 163 |
free fall.
The distance a body released from rest falls in
time $t$ is proportional to the square of the
amount of time it has fallen.
|
\[
s=\frac{1}{2}gt^2
\]
|
| 164 |
free-fall equations (Earth)
|
\[
g=32\frac{\mathrm{ft}}{\mathrm{s}^2}s=16t^2\\
g=9.8\frac{\mathrm{m}}{\mathrm{s}^2}s=4.9t^2
\]
|
| 167 |
If $c$ is the cost of producing $x$ units, then
the
marginal cost
is defined as the cost per
unit when producing $x$ units.
|
\[
\begin{split}
c'\left(x\right)
&=\frac{dc}{dx}\\
&=\lim\limits_{h\rightarrow0}
\frac{c\left(x+h\right)-c\left(x\right)}{h}
\end{split}
\]
|
|
marginal cost (informal)
If $c$ is the cost of producing $x$ units, then the
marginal cost is informally defined as the cost of
producing one more unit. This cost is approximated
by the derivative in the formal definition.
|
\[
\frac{\Delta c}{\Delta x}
=\frac{c\left(x+1\right)-c\left(x\right)}{1}
\]
|
| 167 |
If $r$ is the revenue from producing $x$ units,
then the
marginal revenue
is defined as the revenue per unit when
producing $x$ units.
|
\[
\begin{split}
r^\prime\left(x\right)
&=\frac{dr}{dx}\\
&=\lim\limits_{h\rightarrow0}\frac{r\left(x+h\right)-r\left(x\right)}{h}
\end{split}
\]
|
| 168 |
marginal tax rate.
|
\[
\]
|
| 168 |
marginal revenue (informal)
If $r$ is the revenue from producing $x$ units,
then the marginal revenue is informally defined
as the revenue from producing one more unit.
This revenue is approximated by the derivative
in the formal definition.
|
\[
\frac{\Delta r}{\Delta x}
=\frac{r\left(x+1\right)-r\left(x\right)}{1}
\]
|
| 173 |
product rule for derivatives.
|
\[
\frac{d}{dx}\left[f\left(x\right)g\left(x\right)\right]
=f\left(x\right)g'\left(x\right)
+g\left(x\right)f'\left(x\right)\\
\frac{d}{dx}\left(uv\right)=u\frac{dv}{dx}+v\frac{du}{dx}\\
\frac{d}{dx}\left[\frac{f\left(x\right)}{g\left(x\right)}\right]
=\frac{g\left(x\right)f'\left(x\right)-f\left(x\right)g'\left(x\right)}
{\left[g\left(x\right)\right]^2}
\]
|
| 174 |
quotient rule for derivatives.
|
\[
\frac{d}{dx}\left[\frac{1}{g\left(x\right)}\right]
=-\frac{g'\left(x\right)}
{\left[g\left(x\right)\right]^2}\\
d\left(\frac{1}{v}\right)=-\frac{dv}{v^2}\\
\frac{d}{dx}\left(\frac{u}{v}\right)
=\frac{v\frac{du}{dx}-u\frac{dv}{dx}}{v^2}\\
d\left(\frac{u}{v}\right)=\frac{vdu-udv}{v^2}
\]
|
| 175 |
power rule for negative integers
where $n\lt0\in\Z$
|
\[
\frac{d}{dx}\left(x^n\right)=nx^{n-1}
\]
|
| 180 |
derivative of sine
provided $x$ measured in radians.
|
\[
\frac{d}{dx}\left(\sin{x}\right)=\cos{x}
\]
|
| 181 |
derivative of cosine
provided $x$ measured in radians.
|
\[
\frac{d}{dx}\left(\cos{x}\right)=-\sin{x}
\]
|
| 183 |
derivative of tangent
provided $x$ measured in radians.
|
\[
\frac{d}{dx}\left(\tan{x}\right)=\sec^2{x}
\]
|
| 183 |
derivative of secant
provided $x$ measured in radians.
|
\[
\frac{d}{dx}\left(\sec{x}\right)=\sec{x}\tan{x}
\]
|
| 183 |
derivative of cosecant
provided $x$ measured in radians.
|
\[
\frac{d}{dx}\left(\csc{x}\right)=-\csc{x}\cot{x}
\]
|
| 183 |
derivative of cotangent
provided $x$ measured in radians.
|
\[
\frac{d}{dx}\left(\cot{x}\right)=-\csc^2{x}
\]
|
| 184 |
continuity of trigonometric functions and
calculating limits of algebraic combinations
and composites of trigonometric functions.
where $f$ is a trigonometric function.
The trigonometric functions are continuous on their
domains since they are differentiable. Also,
$\sec$ and $\tan$ are continuous except at nonzero
integer multiples of $\pi/2$ and $\csc$ and $\cot$
are continuous except at integer multiples of $\pi.$
We use this equation to calculate limits of algebraic
combinations and compositions of trig functions.
|
\[
\lim\limits_{x\rightarrow c}f\left(x\right)=f\left(c\right)
\]
|
| 188 |
chain rule
|
\[
\left(f\circ g\right)'\left(x\right)
=f'\left(g\left(x\right)\right)\cdot g'\left(x\right)\\
\frac{dy}{dx}=\frac{dy}{du}\cdot\frac{du}{dx}
\]
|
| 190 |
relationship between parametric and regular derivatives
|
\[
\frac{dy}{dt}=\frac{dy}{dx}\cdot\frac{dx}{dt}
\]
|
| 190 |
parametric formula for the first derivative
to express the derivative as a function of $t:$
|
\[
\frac{dy}{dx}=\frac{dy/dt}{dx/dt}
\]
|
| 191 |
parametric formula for the second derivative
to express the second derivative as a function
of $t$ where $y'=\frac{dy}{dx}$
|
\[
\frac{d^2y}{dx^2}=\frac{dy'/dt}{dx/dt}
\]
|
| 192 |
power chain rule.
Since
\[
\frac{d}{du}\left(u^n\right)=nu^{n-1}
\]
|
\[
\frac{d}{dx}\left(u^n\right)=nu^{n-1}\frac{du}{dx}
\]
|
| 203 |
power rule for rational powers
if $n$ is rational.
|
\[
\frac{d}{dx}x^n=nx^{n-1}
\]
|
Alex Notes