If we remember two things, we can write the equation for the tangent line to f at a given the formula for f and the value a where we want the tangent line to go.
- We can calculate f(a).
- We can calculate f ' (a).
We can write the equation of a line given a point and a slope, so once we have a point and a slope for the tangent line we're all set to go.
We've left out one thing in this discussion of tangent lines: the magic formula. Most textbooks have a magic formula that produces the equation for the tangent line.
We have lots of reasons for leaving this formula out. We don't need it. It takes extra memory that could be better spent remembering the limit definition of the derivative. It can be confusing.
However, for the sake of completeness, we'll show the magic formula. But we're doing it our way.
There's one special case that neither finding the equation of a line nor knowing the magic formula will help with. If f ' (a) is undefined and infinite, then we have a vertical tangent line.
The equation of such a tangent line is x = a like any vertical line. Knowing this will probably be more important for finding derivatives of parametric functions than it will be right now.
We can also work backwards to figure out information about the function given information about its tangent line.