Now we know what the derivative of a function is, and we have some tools to help us calculate them. But what does the derivative mean. Here’s a picture:
In green I’ve drawn a function defined on (at least) the interval of real numbers between and . The specifics of the function don’t matter. In fact having a formula around to fall back on would be detrimental to understanding what’s going on.
In red I’ve drawn the line with equation . This describes a function with two very important properties. First, when we get , so the two functions take the same value there. Second, the derivative everywhere, and in particular . That is, not only do both graphs pass through the same point above , they’re pointing in the same direction. As they pass through the point, the line “touches” the graph of , and we call it the “tangent” line after the Latin tangere: “to touch”.
So the derivative seems to describe the direction of the tangent line to the graph of at the point . Indeed, if we change our input by adding , the tangent line predicts a change in output of . Remember, it’s this simple relation between changes in input and changes in output that makes lines lines. But the graph of the function is not its tangent line, and the function is not the same as the function defined by . How do they differ?
Well, we can subtract them. At , we get a difference of because of how we define the function , so let’s push away to the point . There we find a difference of . But we saw this already in the lead-up to the chain rule! This is the function , where . That is, not only does the difference go to zero — the line and the graph pass through the same point — but it goes fast enough that the difference divided by still goes to zero — the line and the graph point in the same direction.
Let’s try to understand why the tangent line works like this. It’s pretty difficult to draw a tangent line, except in some simple geometric circumstances. So how can we get ahold of it? Well instead of trying to draw a line that touches the graph at that point, let’s imagine drawing one that cuts through at , and also at the nearby point . We’ll call it the “secant” line after the Latin secare: “to cut”. Now along this line we changed our input by and changed our output by . That is, the relationship between inputs and outputs along this secant line is just the difference quotient !
We know that the derivative is the limit of the difference quotient as goes to . In the same way, the tangent line is the limit of the secant lines as we pick our second point closer and closer to — as long as our function is well-behaved. It might happen that the secants don’t approach any one tangent line, in which case our function is not differentiable at that point. In fact, that’s exactly what it means for a function to fail to be differentiable.
So in terms of the graph of a function, the derivative of a function at a point describes the tangent line to the graph of the function through that point. In particular, it gives us the “slope” — the constant relationship between inputs and outputs along the line.