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The Formula of the Tangent Line

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Lesson Summary:

In this lesson we derive the formula for the tangent line given that we know the function and its derivative.

Lesson Inputs:

Lesson:

Let's assume that we have a function $f(x)$ and its derivative <Sign in to see all the formulas>. Our goal here is to find the formula for the tangent line at any arbitrary point $x_0$ .

Since the tangent line is a straight line it has the formula <Sign in to see all the formulas>, our goal is to find the value of $m$ and $b$ .

From the definition of the tangent line we know that <Sign in to see all the formulas> so the hard part is determining $b$ . But we know that at $x_0$ the tangent line has the value <Sign in to see all the formulas>. Therefore, we know that at $x_0$ , <Sign in to see all the formulas>. So now doing some algebra we can quickly find $b$ :

Now we can plug both $m$ and $b$ into the formula <Sign in to see all the formulas> to get the formula for the tangent line:

Finally, with a little algebra we get our final form of the formula for the tangent line: