A function is increasing on an interval if its graph rises as you move to the right, and decreasing if it falls. The derivative tells you which is happening, because f'(x) is exactly the slope of the curve at x.

• If f'(x) > 0, the slope points uphill — f is increasing.
• If f'(x) < 0, the slope points downhill — f is decreasing.
• If f'(x) = 0, the slope is flat for an instant.

Read the sign of the slope

Here is f(x) = x^3 - 3x in bold, with its derivative f'(x) = 3x^2 - 3 below it. Slide the marker and watch the green tangent: when it tilts uphill, f' sits above the x-axis (positive); when it tilts downhill, f' sits below (negative). The curve turns around exactly where f' crosses zero.

The sign line

To find the intervals, first solve f'(x) = 0. For f'(x) = 3x^2 - 3 = 3(x-1)(x+1) that gives x = -1 and x = 1. These split the line into three pieces; test one point in each to read off the sign of f':

So f increases, then decreases, then increases — exactly the shape you saw above. The graphic below makes the sign line interactive: drag the test point and see whether f' there is positive or negative.

Watch it on Khan Academy