We've just seen that a flat line — a constant function — has slope 0 everywhere. Now let the line tilt. The simplest sloped function there is:

Whatever goes in comes straight back out. Its graph is the line through the origin at 45^\circ: for every step you take to the right, the output rises by exactly the same amount.

A line through the origin: y = mx

Before we zoom in on y = x, look at the whole family it belongs to. Any straight line through the origin can be written

where m is the slope. Slope is \dfrac{\text{rise}}{\text{run}}: take a run of 1 to the right and the line rises by exactly m, so \dfrac{\text{rise}}{\text{run}} = \dfrac{m}{1} = m — and it's the same at every point on the line. Drag the point along the line, and move the m slider to tilt it: the run stays 1 while the rise tracks m.

Now set m = 1. The line is our y = x: a run of 1 gives a rise of 1, so its slope is exactly 1 — the same everywhere. That constant slope is the derivative we're after.

The difference quotient confirms it

Drop f(x) = x into the difference quotient. The output at x + h is just x + h:

The x's cancel, leaving \dfrac{h}{h} = 1 — and that's already 1 before we shrink h, so the limit is 1 too. Our second rule:

See it together

On the left, the line f(x) = x climbing at a steady 45^\circ. On the right, its derivative f'(x) = 1 — a flat line at height 1, because the slope is the constant 1 everywhere.