How steep, at every point

Derivatives, visually

The derivative is a new function that tells you, at every x, how steep the old one is. Height and steepness are different things. Drag the point along the curve and watch the beam and the slope graph.

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at x = 3: height 3.25, slope 0.5

Point x 3

f(3) = 3.25: the height of the curve
f′(3) = 0.5: how steep it is
f′(x) = 0.5x − 1, so it is level at x = 2

The beam climbs a little here, so the slope is positive. At x = 2 the beam is level.

Idea 1 · From average slope to slope at a point

From average slope to the slope at a point

The slope between two points on a curve is rise over run: the average rate of change, drawn as a line called a secant.

Slide the second point toward the first and the line settles on the tangent, the beam that touches the curve at one spot. Its slope is the derivative at that point.

Aha!

Bring the second point closer and closer: the line settles on the beam that touches the curve at one spot.

Idea 2 · The slope graph

Draw the slope graph

At every x the curve has a steepness. Plot those steepnesses and you get the derivative function. For y = x² it is the straight line 2x.

Plot slopes, not heights: plotting heights again is the usual mistake. The new graph answers how steep, not how high.

Aha!

The derivative is a new function that tells, at every x, how steep the old one is.

Idea 3 · Peaks and valleys

Peaks and valleys: where the beam is level

Where the curve climbs, the slope is positive. Where it falls, the slope is negative. At a peak or a valley the beam is level: the slope is 0.

Be careful: a slope of 0 is not always a peak. Check what the slope does on each side. Reading the sign of f instead of the sign of f′ is the usual slip.

Aha!

Plus climbs, minus falls, and level means a possible peak or valley.

Idea 4 · The power rule

The power rule: bring the power down

To find the slope of xⁿ, bring the power down in front and lower the power by one: x³ gives 3x², x² gives 2x, and a plain number gives 0.

For sums, do each piece on its own: 5x² − 3x + 7 gives 10x − 3. Forgetting to lower the power is the usual mistake. Check it: at x = 1 the slope of x³ is 3, and the beam agrees.

Aha!

Bring the power down, then lower the power by one.

Quick answers

Questions about derivatives

What is a derivative in simple words?

The slope of a curve at a point. As a function, f′(x) tells you how steep the curve is at every x.

What is the difference between f(x) and f′(x)?

f(x) is the height of the curve at x. f′(x) is how steep the curve is at x. A curve can be high and flat, or low and steep.

What does it mean when the derivative is zero?

The tangent beam is level, so the curve may have a peak or a valley there. Check the slope on each side to be sure: some points with slope 0 are neither.

How do you use the power rule?

Bring the power down in front and lower the power by one. For x⁵ the derivative is 5x⁴. For a sum, differentiate each term: 3x² + 4x has derivative 6x + 4.

Now play it

Try it in Pip: Math Adventure

World 10 of Pip: Math Adventure is about derivatives. You drive the hill and aim the beam, and learn to find peaks, speeds and the best value in a real problem.

It is a math game for iPhone, coming soon to the App Store. Join the list and we will tell you the day Pip is out.

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