What a derivative is

The speedometer problem

Speed is distance divided by time. That works fine for a whole trip: 120 miles in 2 hours is 60 miles per hour on average. But a speedometer shows your speed right now, at a single instant. An instant has zero time, and you can't divide by zero. So what does "speed right now" even mean?

Shrink the gap

Take a ball that rolls so that after t seconds it has gone s(t) = t² meters. Find its average speed over a gap, starting at 3 seconds:

| Gap | Distance | Average speed | |---|---|---| | 3 to 4 s | 9 m to 16 m | (16 − 9) ÷ 1 = 7 m/s | | 3 to 3.1 s | 9 m to 9.61 m | (9.61 − 9) ÷ 0.1 = 6.1 m/s | | 3 to 3.01 s | 9 m to 9.0601 m | (9.0601 − 9) ÷ 0.01 = 6.01 m/s |

The averages are 7, 6.1, 6.01: they close in on 6. So at exactly 3 seconds, the ball's speed is 6 m/s. We never divided by zero; we watched where the averages were heading.

The derivative and its notation

That instant rate of change is called the derivative. For a function y = f(x), you'll see it written two ways:

• f′(x), read "f prime of x"
• dy/dx, read "d y d x": a tiny change in y over a tiny change in x

For the ball, the position function is s(t), so its derivative is written s′(t), and s′(3) = 6.

The slope at one point

On a graph, an average rate of change is the slope of a secant line, a line through two points on the curve. As the second point slides toward the first, the secant line tilts and settles into the tangent line, which just touches the curve at that point. The derivative is the slope of that tangent line: rise over run at one point. On the graph of s(t) = t², the tangent line at t = 3 has slope 6.

Try it yourself

Use the same ball, s(t) = t², but at t = 1. Work out the average speed from 1 to 2 seconds, from 1 to 1.1 seconds and from 1 to 1.01 seconds. What number do the averages close in on? (Answers: 3, 2.1 and 2.01 m/s, closing in on 2 m/s, so s′(1) = 2.)

Good to know

• The derivative is the instantaneous rate of change, the limit of average rates of change over shrinking intervals, and graphically the slope of the tangent line at a point.
• The ball is an idealized example with position s(t) = t² meters; "speed" here is the size of its velocity, since it moves in one direction.
• The averages over a gap of length h starting at 3 are exactly 6 + h (h = 1, 0.1, 0.01 gives 7, 6.1, 6.01), which is why they head to 6.
• f′(x) and dy/dx mean the same thing for a function y = f(x). The formal limit definition of the derivative is the next lesson.

Answer to the comment question

y = 4x + 1. What's its derivative? dy/dx = 4. The graph of y = 4x + 1 is a straight line with slope 4, and a straight line has the same slope at every point, so the slope at any one point is 4.