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\documentclass[12pt]{article}
\title{Definition Homework}
\author{Joshua Garrett}
\usepackage {amsmath}
\usepackage {amssymb}
\newtheorem{thm}{Definition}
\newtheorem{example}{Example}
\begin{document}
\title{Definition Homework}
\author{Joshua Garrett}
\textsc{\LARGE Definition of the Derivative}\\
[1.5cm]
\indent
\textsc{\Large Joshua Garrett}\\
[0.5cm]
\indent
The derivative is a useful definiton because it gives the tangent line to the given equation. The derivative is a basic term that is applied in most advanced mathematical concepts such as Abstract Algebra, Differential Equations, Electrodynamics, and other advance courses.
\begin{thm}[The Formal Definition of Derivative]\label{Derivative}
The derivative is defined as the computation of the slope of a tangent line, the instantaneous rate of change of a continuous function, and the instantaneous velocity of an object. The derivative of $f(x)$ with respect to x is the function $f^\prime$ and is defined as,
\begin{equation}
f^\prime (x) =
\lim_{h\rightarrow0} \frac{f(x+h) - f(x)}{h}
\end{equation}
\end{thm}
\begin{thm}[The Informal Definition of Derivative]
The derivative of a continuous function is the rate of change of a function at a given input. The derivative of a function can also be denoted as
\begin{equation}
{(f^{n} (x)})^{\prime} = n\cdot {f^{n-1}(x)}
\end{equation}
\end{thm}
\begin{example}[Solution Attainable]
Find the derivative of the function using the limit definition:
\begin{equation}
f(x) = x^2+7
\end{equation}
Solution:
\begin{eqnarray*}
f^\prime (x) &=&
\lim_{h\rightarrow0} \frac{((x+h)^2+7) - (x^2+7)}{h} \\
&=& \lim_{h\rightarrow0} \frac{x^2 + 2xh + h^2 +7-x^2-7)}{h} \\
&=& \lim_{h\rightarrow0} \frac{2xh+h^2}{h} \\
&=& \lim_{h\rightarrow0} \frac{h(2x+h)}{h} \\
&=& \lim_{h\rightarrow0} 2x+h \\
&=& 2x
\end{eqnarray*}
\end{example}
\begin{example}[Solution Unattainable]
Find the derivative of the function using the limit definition:
\begin{equation}
f(x)=\left\{
\begin{array}{c l}
x^2 & x>0\\
-x^3 & x<0
\end{array}\right.
\end{equation}
Solution:
Since the function above is a piece wise function, there is a point at x = 0 where the function has a hole. Since the function is discontinues, the derivative does not exist.
\end{example}
\end{document}