Calculus I MATH 131 Fall 2025

Start with an example. Let \(f(x) = x^2 + 1\text{.}\) Find \(f'(3)\text{.}\) Then, what is \(f'(4)\text{?}\) Can we also find \(f'(42)\text{?}\)

In fact, we can find \(f'(a)\) for any number \(a\text{.}\) One way to think of this is that we are creating a rule to help us find a lot of different derivatives. We are giving a recipe. No matter what \(a\) is, we can always find \(f'(a)\) by simply...

But really we are doing more. At any point in the domain of \(f\text{,}\) we can find a derivative. That derivative changes when \(a\) changes. So there is actually a function of derivatives.

We can describe a function in multiple ways. A formula. A table. A graph.

Look at a graph of a function. At any input, we can ask: what is the slope of the line tangent to the graph? We can then collect these answers in a table. That is describing the derivative function! We can also graph these values to get a graph of \(f'(x)\)