We learned to find the derivative at a single point, f'(a). But there is nothing special about a — we can do it at every point. Let the input roam and the answer becomes a brand-new function.

Replace the fixed a with a variable x. The result is the derivative function f'(x):

Feed f' any input x and it hands back the slope of f at that x. The derivative is a function whose output is "the steepness of f here".

From x^2 to 2x

We already simplified the difference quotient for f(x) = x^2 to 2x + h. Taking the limit as h \to 0 keeps only the part with no h:

Notice what f'(x) = 2x tells us at a glance: the slope is negative for x < 0 (the curve falls), zero at x = 0 (the flat bottom), and positive and growing for x > 0 (the curve climbs ever more steeply). One tidy formula captures every tangent slope at once.

Read f' off the slope of f

Here is the key picture. The top curve is f(x) = x^2; the bottom curve is its derivative f'(x) = 2x. Slide x: the dot on top shows the tangent slope there, and the dot on the bottom plots that very slope as a height. Trace the bottom dot and you draw f'.

Watch the slope become a curve

Press play: as the tangent point sweeps across f(x) = x^2, its slope value is dropped below to trace out the line f'(x) = 2x.

Watch it explained

Sal Khan derives the derivative of f(x) = x^2 for any x — turning the slope into a function.