Title: Math 2250 Facts to Memorize
Tags: Derivative Trig Rules Concepts Exponential Polynomial

Q: [latex]$\frac{d}{dx} \sin x$[/latex]
A: [latex]$\cos x$[/latex]

Q: [latex]$\frac{d}{dx} \cos x$[/latex]
A: [latex]$-\sin x$[/latex]

Q: [latex]$\frac{d}{dt} \tan x$[/latex]
A: [latex]$\sec^2 x$[/latex]

Q: [latex]$\frac{d}{dx} \cot x$[/latex]
A: [latex]$-\csc^2 x$[/latex]

Q: [latex]$\frac{d}{dx} \sec x$[/latex]
A: [latex]$\sec x \tan x$[/latex]

Q: [latex]$\frac{d}{dx} \csc x$[/latex]
A: [latex]$-\csc x \cot x$[/latex]

Q: [latex]$\frac{d}{dx} \arcsin x$[/latex]
A: [latex]$\frac{1}{\sqrt{1 - x^2}}$[/latex]

Q: [latex]$\frac{d}{dx} \arccos x$[/latex]
A: [latex]$\frac{-1}{\sqrt{1 - x^2}}$[/latex]

Q: [latex]$\frac{d}{dx} \arctan x$[/latex]
A: [latex]$\frac{1}{1+x^2}$[/latex]

Q: [latex]$\frac{d}{dx} \operatorname{arccot} x$[/latex]
A: [latex]$\frac{-1}{1 + x^2}$[/latex]

Q: [latex]$\frac{d}{dx} x^n$[/latex]
A: [latex]$n x^{n-1}$[/latex]

Q: [latex]$\frac{d}{dx} e^x$[/latex]
A: [latex]$e^x$[/latex]

Q: [latex]$\frac{d}{dx} a^x$[/latex]
A: [latex]$a^x \ln a$[/latex]

Q: [latex]$\frac{d}{dx} \ln x$[/latex]
A: [latex]$\frac{1}{x}$[/latex]

Q: [latex]$\frac{d}{dx} (f(x) + g(x))$[/latex]
A: [latex]$f'(x) + g'(x)$[/latex]

Q: [latex]$\frac{d}{dx} k f(x)$[/latex]
A: [latex]$k f'(x)$[/latex]

Q: [latex]$\frac{d}{dx} f(x) \cdot g(x)$[/latex]
A: [latex]$f'(x) g(x) + f(x) g'(x)$[/latex] (product rule)

Q: [latex]$\frac{d}{dx} f(g(x))$[/latex]
A: [latex]$f'(g(x)) g'(x)$[/latex] (chain rule)

Q: [latex] The derivative of \textbf{velocity} $v(t)$ with respect to \textbf{time} $t$ is \dots [/latex]
A: [latex] \textbf{acceleration} $a(t)$.[/latex]

Q: [latex] A \textbf{critical point} of a function $f(x)$ is a point where \dots [/latex]
A: [latex]$f'(x) = 0$ OR $f'(x)$ does not exist (a corner or asymptote).[/latex]

Q: [latex] The maximum and minimum values of $f(x)$ on a \textbf{closed} interval $[a,b]$ are located at \dots [/latex]
A: [latex]$a$, $b$ \textbf{or} a critical point of $f(x)$.[/latex]

Q: [latex] $$\frac{d}{dx} \frac{f(x)}{g(x)} = $$[/latex]
A: [latex]$$\frac{f'(x) g(x) - f(x) g'(x)}{g^2(x)}$$[/latex]

Q: [latex] The \textbf{quadratic} approximation to $f(x+\delta x)$ is [/latex]
A: [latex] $f(x + \delta x) \simeq f(x) + f'(x) \delta x + \frac{f''(x)}{2} (\delta x)^2$ [/latex]

Q: [latex]
(L'H\^ospital's Rule). If $f(a) = g(a) = 0$ and $f'(a)$ and $g'(a)$ exist and $g'(a)$ is not zero, then
\begin{equation*}
\lim_{x \rightarrow a} \frac{f(a)}{g(a)} =
\end{equation*}
[/latex]
A: [latex]
\begin{equation*}\lim_{x \rightarrow a} \frac{f(a)}{g(a)} = \lim_{x \rightarrow a} \frac{f'(a)}{g'(a)}
\end{equation*}
[/latex]