The derivative, seen as a secant that turns into a tangent

How to use this simulation

1. Pick a function from “Function f(x)”: x², x³, sin x, eˣ or 1/x. The thick curve on top is y = f(x). The black dot P is the point we measure the steepness at. Drag along the top graph, or use “x of point P”, to move it.
2. The orange dot Q sits a gap h away from P, and the orange line through both is the secant. At the start h = 2, so Q is far away and the secant is a rough average slope, nowhere near the curve’s real steepness at P.
3. Press play. The gap h shrinks smoothly, Q slides along the curve towards P, and the secant swings round. When h reaches 0 the line turns green: it is the tangent, the line that just touches the curve at P.
4. Watch the readings. “Secant slope” is (f(x + h) − f(x)) / h. “Exact derivative” is the true f′(x). As h shrinks, “Difference between them” falls towards zero. Pause and drag the h slider yourself to see it at any size.
5. Look at the lower graph. It shows y = f′(x), the derivative function. The green dot marks f′ at P and the hollow orange ring marks the current secant slope. The ring slides into the dot as h goes to 0.
6. Drag P to a new place and repeat. On x² the green dot climbs a straight line; on sin x it follows a cosine wave. The tangent equation in the readings updates for every position.

How steep is a curve?

A straight line has one slope everywhere: rise over run, the same number from start to finish. A curve is trickier. A ball thrown in the air climbs steeply, flattens at the top, then drops. Its steepness changes from one moment to the next. So what does “the slope at this exact point” even mean?

The trick is to zoom in. Zoom in far enough on any smooth curve and it looks like a straight line. The slope of that tiny straight stretch is the slope of the curve at that point, and the straight line itself is called the tangent. The derivative is the machine that gives you that number.

From a secant to a tangent

We cannot find a slope from one point alone, we need two. So we cheat. Take P on the curve and a second point Q a little way along. The line through both is called a secant, and its slope is easy: rise over run.

Now let Q slide towards P. The secant swings and the slope changes. Here is x² at x = 1, where P = (1, 1). The first column is the gap h, the second is the secant slope.

• With h = 1 the secant slope is 3.0000. That is the average slope over a whole unit, so it is far from the truth.
• With h = 0.1 it is 2.1000, and with h = 0.001 it is 2.0010.
• The numbers are marching towards 2. That target is the derivative of x² at x = 1.

The definition, written down

The slope of the secant through x and x + h is (f(x + h) − f(x)) / h. The derivative is what that settles on as h shrinks to zero. That “settles on” has a precise name, a limit.

We never actually put h = 0, because 0 ÷ 0 means nothing. We only ask what the ratio gets closer and closer to. In the simulation the last step jumps to h = 0 just to draw the finished tangent.

The derivative is a graph too

Move P and the tangent slope changes. Write down the slope for every x and you get a new function, f′(x). That is the lower graph in the simulation. Whenever the top curve rises, the lower graph sits above zero. Where the top curve has a peak or a trough, the lower graph crosses zero. Where the top curve falls, the lower graph is negative.

For x² the table shows the pattern: the height and the slope at five points. The slope is exactly twice x.

The five functions in the simulation

Each of these has a known derivative. The table lists it, and the last column is the slope at x = 1, so you can check it against the simulation by setting “x of point P” to 1.

• x² has a slope that grows steadily: 2x. It is flat at 0, rising on the right, falling on the left.
• x³ has slope 3x², which is never negative. The curve never goes downhill, but it flattens for a moment at 0.
• sin x has slope cos x. The derivative graph is the same wave shifted a quarter turn.
• eˣ is the one function whose slope equals its height, everywhere. The two graphs are identical.
• 1/x has slope −1/x², always negative. The curve falls on both sides of the break at 0.

Try it in the simulation

Six quick experiments, a minute each.

• Choose x². Press play and watch h shrink. Note the secant slope at the end and compare it with 2x at the P you chose.
• Pause, drag P to x = 0 on x². The tangent is flat. What does the lower graph read there?
• Choose eˣ and look at the two graphs. Can you tell which is which without the labels?
• Choose sin x. Drag P to the top of the wave. The tangent goes flat. Now drag to where the curve crosses the axis. Is the tangent steepest here?
• Choose 1/x. Drag P close to 0 on the right, then on the left. Why does the slope blow up?
• Choose x³, set h to 2, and drag P around. The secant overshoots at large h. At what size of h does it start to look like the tangent?

Solved problems

Eight common questions. Every number below is computed with the same functions as the simulation.

Common mistakes

Four slips that cost marks every year.

• Putting h = 0 straight into the fraction. You get 0 ÷ 0. Expand and simplify first, then let h shrink.
• Confusing the secant with the tangent. The secant joins two points on the curve. The tangent touches at one point.
• Reading the derivative as the height. f′(a) is the slope at a, not the value f(a).
• Forgetting the point when writing the tangent: use y = f′(a)(x − a) + f(a), not just y = f′(a)x.

Where derivatives are used

Speed is the derivative of position. Acceleration is the derivative of speed. The rate at which a population grows, a tank empties, or a drug leaves the bloodstream is a derivative. Economists use it for marginal cost, engineers for how fast a signal rises, and machine learning trains by following slopes downhill.

Exam corner

Typical SSC higher-math, HSC and first-year questions on this topic.

• Find f′(x) by the definition (first principles), for x², x³ or 1/x.
• Write the equation of the tangent at a given point.
• Find the point where the tangent is parallel to a given line, or horizontal.
• Read from a graph where the derivative is positive, zero or negative.

Revision summary

Slope of a curve = slope of its tangent. Secant slope = (f(x + h) − f(x)) / h. Let h go to 0 and the secant becomes the tangent. The limit is the derivative f′(x). Tangent line: y = f′(a)(x − a) + f(a).

Frequently asked questions