One of the more popular applications of a derivative is to actually find the equation of the tangent linethat we keep referring to.

The reason this is such a common question, is because the information you use to create the equation of *
any
* line, a point, $\left({\textcolor[rgb]{}{x}}_{\textcolor[rgb]{}{1}}\textcolor[rgb]{}{,}{\textcolor[rgb]{}{y}}_{\textcolor[rgb]{}{1}}\right)$, and a slope, **
m
**, are always going to be something you are always able to easily find. Once you have a point, $\left({\textcolor[rgb]{}{x}}_{\textcolor[rgb]{}{1}}\textcolor[rgb]{}{,}{\textcolor[rgb]{}{y}}_{\textcolor[rgb]{}{1}}\right)$, and a slope,

**
Meaning:
****
Tangent Line
**

Tangent line means a *
line
* (*
linear equation
*) that touches your actual equation, $\textcolor[rgb]{}{f}\textcolor[rgb]{}{\left(}\textcolor[rgb]{}{x}\textcolor[rgb]{}{\right)}$, at just one point, $\left(\textcolor[rgb]{}{x}\textcolor[rgb]{}{,}\textcolor[rgb]{}{y}\right)$, one instance. That tangent point is the intersection of your tangent line and the original equation, $\textcolor[rgb]{}{f}\textcolor[rgb]{}{\left(}\textcolor[rgb]{}{x}\textcolor[rgb]{}{\right)}$.

**
Note
**:

–
This is *
not
**
tangent
* like the *
sine
*, *
cosine
*, *
tangent
* that you learned in your trig class. They are *
not related
* to each other at all.

**
Definition:
****
Point
****
–
****
slope
****
form of a line.
**

$\textcolor[rgb]{}{y}\textcolor[rgb]{}{=}\textcolor[rgb]{}{m}\textcolor[rgb]{}{(}\textcolor[rgb]{}{x}\textcolor[rgb]{}{\u2013}{\textcolor[rgb]{}{x}}_{\textcolor[rgb]{}{1}}\textcolor[rgb]{}{)}\textcolor[rgb]{}{+}{\textcolor[rgb]{}{y}}_{\textcolor[rgb]{}{1}}$

**
Note
**:

–
I know that this version of the Point–Slope Form of a Line is probably slightly different than the one in your textbook or the one your instructor is using. I prefer this version because it takes one step of algebra out of the process by moving the ${\textcolor[rgb]{}{y}}_{\textcolor[rgb]{}{1}}$to the right of the equals sign to start with. I **
guarantee
** you that it is the exact same thing, and that there is

–
If you would like to stick with the more common version, and do the algebra on your own, then here is a more standard point–slope form for a *
line
*.

$\textcolor[rgb]{}{y}\textcolor[rgb]{}{\u2013}{\textcolor[rgb]{}{y}}_{\textcolor[rgb]{}{1}}\textcolor[rgb]{}{=}\textcolor[rgb]{}{m}\textcolor[rgb]{}{(}\textcolor[rgb]{}{x}\textcolor[rgb]{}{\u2013}{\textcolor[rgb]{}{x}}_{\textcolor[rgb]{}{1}}\textcolor[rgb]{}{)}$

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