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# Finding tangent line equations using the formal definition of a limit

This structured practice takes you through three examples of finding the equation of the line tangent to a curve at a specific point.
We can calculate the​ slope of a tangent line using the definition of the derivative of a function $f$ at $x=c$ (provided that limit exists):
$\underset{h\to 0}{lim}\frac{f\left(c+h\right)-f\left(c\right)}{h}$
​Once we've got the slope, we can ​find the equation of the line. This article walks through three examples.

## Example 1: Finding the equation of the line tangent to the graph of $f\left(x\right)={x}^{2}$‍  at $x=3$‍

Step 1
What's an expression for the derivative of $f\left(x\right)={x}^{2}$ at $x=3$?

Step 2
Evaluate the correct limit from the previous step.
${f}^{\prime }\left(3\right)=$

${f}^{\prime }\left(3\right)$ gives us the slope of the tangent line. To find the complete equation, we need a point the line goes through.
Usually, that point will be the point where the tangent line touches the graph of $f$.
Step 3
What is the point we should use for the equation of the line?
$\left($
$,$
$\right)$

Step 4
Complete the equation of the line tangent to the graph of $f\left(x\right)={x}^{2}$ at $x=3$.
$y=$

And we're done! Using the definition of the derivative, we were able to find the equation for the line tangent to the graph of $f\left(x\right)={x}^{2}$ at $x=3$.

## Example 2: Finding the equation of the line tangent to the graph of $g\left(x\right)={x}^{3}$‍  at $x=-1$‍

Step 1
${g}^{\prime }\left(-1\right)=?$