Method: Equation of a Tangent Line

The process for finding the equation of the tangent line is really the same process you used to find the equation of any line in your algebra course. The only difference is how you go about finding the slope of the line , the m . In calculus you use the derivative equation, $f^{\prime }\left(x\right)$, to find the slope of the tangent line .

Step 1: Find the tangent point, $\left(x_1,y_1\right)$.

This is the point shared by both the tangent line and the original equation, $f\left(x\right)$, their point of intersection. Sometimes you will be given the tangent point, $\left(x_1,y_1\right)$, in the language of the problem, and sometimes you will need to find the tangent point yourself. It is common to just be given the x-value where they would like you to find the tangent line at. You will want to take your given x-value , $x_1$, and plug it into the original equation, $f\left(x\right)$, to find the y-value that goes with it, $f\left(x_1\right)=y_1$. Remember you find y-values by plugging in x-values .

Step 2: Find the slope , m , of the tangent line.

You find the slope , m , of the tangent line using the derivative, $f^{\prime }\left(x\right)$. In calculus the derivative equation, $f^{\prime }\left(x\right)$, is our new equation for determining the slope .

You will evaluate the derivative, $f^{\prime }\left(x\right)$, at the same x-value that you used to find the tangent point in Step 1. This is the x-value from your $\left(x_1,y_1\right)$.

$f^{\prime }\left(x_1\right)=m=\mathit{slope}$

Step 3: Find the equation of the tangent line.

You take the tangent point, $\left(x_1,y_1\right)$, from Step 1 .

You take the slope, $f^{\prime }\left(x_1\right)=m=\mathit{slope}$, from Step 2.

You plug those values into the point-slope form of a line, $y=m(x–x_1)+y_1$, to get the final result for the equation of the tangent line.

Unless requested to simplify to the more common slope-intercept form of a line, y=mx+b , you can always leave your answer in point-slope form.

Do extra work only when requested.