Title: Math 2250 Chapter 3 (Derivatives)
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]A derivative $\frac{d}{dt} f(t)$ with respect to a variable $t$ representing time is \dots [/latex]
A: [latex] the \emph{rate of change} of $f$ with respect to time. [/latex]

Q: [latex]The derivative of \textbf{position} $p(t)$ with respect to \textbf{time} $t$ is \dots [/latex]
A: [latex] \textbf{velocity} $v(t)$.[/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{definition} of the derivative $\frac{d}{dx} f(x)$ is [/latex]
A: [latex] $$\frac{d}{dx} f(x) = \lim_{h \rightarrow 0} \frac{f(x+h) - f(x)}{h}$$ [/latex]

Q: [latex] The \textbf{linear} approximation to $f(x+h)$ is [/latex]
A: [latex]$f(x + h) \simeq f(x) + f'(x) h$[/latex].

Q: [latex] The \textbf{slope of the tangent line} to $f(x)$ at $x=a$ is [/latex]
A: [latex] $f'(a)$ [/latex]

Q: [latex]
The derivative of \textbf{velocity} $v(t)$ with respect to time is
[/latex]
A: [latex]
\textbf{acceleration} $a(t)$.
\begin{equation*}
v'(t) = a(t).
\end{equation*}
[/latex]

Q: [latex]
The acceleration due to gravity is approximately
[/latex]
A: [latex]
32 feet per second per second ($32 \operatorname{ft}/\operatorname{sec}^2$)
or
9.8 meters per second per second ($9.8 \operatorname{m}/\operatorname{sec}^2$)
This acceleration is in the \textbf{downward} direction, so
$a(t) = -32$ or $a(t) = -9.8$ if the coordinate system is pointing straight up.
[/latex]

Q: [latex]
The difference between \emph{velocity} and \emph{speed} is
[/latex]
A: [latex]
Speed is the absolute value $|v(t)|$ of velocity $v(t)$.
[/latex]

Q: [latex]
To find $\frac{dy}{dx}$ given an equation like
\begin{equation*} y^2 = x^2 + \sin xy
\end{equation*}
by \emph{implicit differentiation}
[/latex]
A: [latex]
\begin{enumerate}
\item Differentiate both sides with respect to $x$, writing $\frac{dy}{dx}$ for the derivative of $y$ with respect to $x$, and using the chain rule to differentiate functions of $y$. For example, the derivative of $y^2$ with respect to $x$ is $2 y \frac{dy}{dx}$.
\item Use algebra to solve for $\frac{dy}{dx}$ in terms of $x$ and $y$.
\end{enumerate}
[/latex]

Q: [latex]
The \emph{inverse function} of $f(x)$, written $f^{-1}(x)$ is
[/latex]
A: [latex]
the function so that $f(f^{-1}(x)) = x$.
Examples
\begin{enumerate}
\item $\operatorname{arcsin}(x)$ is the inverse function for $\sin(x)$, so $\sin(\arcsin(x)) = x$.
\item $\ln x$ is the inverse function for $e^x$, so $e^{\ln x} = x$.
\end{enumerate}
[/latex]

Q: [latex]
The derivative of the inverse function $f^{-1}(x)$ for $f(x)$ at $x = b$ can be written in terms of $f$ as
[/latex]
A: [latex]
\begin{equation*}
\frac{d}{dx} f^{-1}(x) = \frac{1}{f'(f^{-1}(b))}
\end{equation*}
[/latex]

Q: [latex]
If we know $f(a)$ and $f'(a)$, we can approximate $f(x)$ for $x$ near $a$ by
[/latex]
A: [latex]
\begin{equation*}
f(x) \simeq f(a) + f'(a) (x - a).
\end{equation*}
[/latex]

Q: [latex]
If we know $f(a)$, $f'(a)$ and $f''(a)$, we can approximate $f(x)$ for $x$ near $a$ by
[/latex]
A: [latex]
\begin{equation*}
f(x) \simeq f(a) + f'(a) (x - a) + \frac{f''(a)}{2} (x-a)^2.
\end{equation*}
[/latex]

Q: [latex]The logarithmic differentiation formula is
$$
f'(x) =
$$
[/latex]
A: [latex]
$$
f'(x) = f(x) \, \frac{d}{dx} \ln f(x).
$$
This is usually used when the laws of logarithms allow you to simplify $\ln f(x)$ \emph{before} taking the derivative, as in
$$
f(x) = x^{\sin x}
$$
where
$$
\ln f(x) = \ln \left( x^{\sin x} \right) = (\sin x) (\ln x).
$$
[/latex]

Q: [latex]The derivative
$$
\frac{d}{dx} \ln f(x) =
$$
[/latex]
A: [$$]
\frac{d}{dx} \ln f(x) = \frac{f'(x)}{f(x)}.
[/$$]

Q: [latex]
The derivative
$$
\frac{d}{dx} \log_a x =
$$
[/latex]
A: [$$]
\frac{d}{dx} \log_a x = \frac{1}{x \ln a}.
[/$$]