Speak Maths · Last school years

How fast: derivatives

A derivative is a rate of change at one moment, like the needle on a speedometer. Six signs say it.

Draft: written by an AI helper and not yet checked by a teacher.

The alphabet

Read each sign aloud, the way it is written under it.

$Δ x$ Say: delta xMeans: a change in x: the end value minus the start value
$\frac{Δ y}{Δ x}$ Say: delta y over delta xMeans: the average rate of change over a stretch: change in y per change in x
$\frac{d y}{d x}$ Say: d y by d xMeans: the rate of change of y at one single moment
$f^{'} (x)$ Say: f prime of xMeans: the derivative of f at x, another way to write d f by d x
$\frac{d ^{2} y}{d x ^{2}}$ Say: d two y by d x squaredMeans: the rate at which the rate of change is itself changing
$x ˙$ Say: x dotMeans: the derivative of x with respect to time

The story: The speedometer

Subtitles:

Tap a line to see its words.

$x = f (t) = 2 t^{2}$ A car pulls away. After t seconds it has gone x metres, where x is f of t, equal to 2 t squared.
$Δ t = 3 - 1 = 2$ Look at the stretch from 1 second to 3 seconds. It lasts 2 seconds.
$Δ x = 18 - 2 = 16$ In that time the car goes from 2 metres to 18 metres: 16 metres further.
$\frac{Δ x}{Δ t} = \frac{16}{2} = 8$ Average speed: 16 metres in 2 seconds, which is 8 metres per second.
$\frac{Δ x}{Δ t} = \frac{2.42 - 2}{0.1} = 4.2$ Shrink the stretch to 1 to 1.1 seconds: the car goes 0.42 metres in 0.1 seconds, which is 4.2 metres per second.
$\frac{d x}{d t} = f^{'} (t) = 4 t$ Shrink it to nothing and you get the derivative, d x by d t, which is f prime of t, equal to 4 t.
$x ˙ (1) = 4$ At 1 second the speedometer needle shows 4 metres per second.
$x ˙ (5) = 20$ At 5 seconds it shows 20 metres per second.
$\frac{d ^{2} x}{d t ^{2}} = 4$ The speed rises by 4 metres per second every second: the second derivative is 4.

In words this story is 704 letters long; in symbols, 98.

Read it

What does it say?

$\frac{Δ y}{Δ x}$

$\frac{d y}{d x}$

$\frac{d ^{2} y}{d x ^{2}}$

$x ˙$

$Δ T = - 3$

$f^{'} (4) = 0$

$\frac{d V}{d t} = 2$

$\frac{d ^{2} h}{d t ^{2}} = - 10$

$\frac{Δ y}{Δ x} = \frac{f ( 5 ) - f ( 2 )}{5 - 2}$

$V ˙ = - 0.5$

$f^{'} (x) > 0$

$\frac{d y}{d x}_{x = 3} = 7$

Write it

Which notation says this?

The derivative of f at x.

The average rate of change of y over a stretch of x.

The second derivative of y with respect to x.

The change in temperature: the end value minus the start value.

The speed of a runner: the derivative of position with respect to time.

The slope of f at 3 is minus 2.

Acceleration: the second derivative of position with respect to time.

Right now the position is growing at 3 units per second, written with the dot.

The average rate of change of f between 1 and 4.

y is x cubed, so d y by d x is 3 x squared.

The ball is still rising but slowing down: its speed is positive and its second derivative is negative.

The bath drains at half a litre per minute: the time-derivative of the volume is minus one half.

Build it

Tap the tiles in order. One tile does not belong.

f prime of 2 equals 6.

Delta y over delta x equals 5.

d x by d t equals 4 t.

Delta y equals f of 5 minus f of 2.

The second derivative of x with respect to t is minus 10.

x dot at 3 equals zero: the cart stops for a moment.

Challenges

Creative puzzles that use every sign of this chapter. Take your time; there is no clock.

1 A walk there and back

A walker's position is x = t² − 4t. Judge each sentence about the walk: true or false?

	$x ˙ > 0$	$x ˙ = 0$	$x ˙ < 0$	$x < 0$
$t = 0$			✓
$t = 1$			✓	✓
$t = 2$		✓		✓
$t = 3$	✓			✓
$t = 4$	✓

A tick means the column's claim holds at that moment, for x = t² − 4t. The speed is ẋ = 2t − 4.

$x ˙ (3) > 0$
$x ˙ (1) = 0$
$x ˙ (2) = 0$
$\frac{d ^{2} x}{d t ^{2}} < 0$
$x (4) - x (0) = Δ x = 0$
$\frac{x ( 2 ) - x ( 0 )}{2 - 0} = - 4$

2 Change in every dress

Pair each notation with what it says.

Tap a line, then its meaning.

3 Average speed

Position is x = t², from t = 1 to t = 3. This should give the average speed. One tile is wrong. Find it, then fix it.

Tap the tile that is wrong.

4 The second time round

Differentiate y = x³ twice. One tile is wrong. Find it, then fix it.

Tap the tile that is wrong.

5 Slope at 3

What is the derivative of x² at x = 3?

$f (x) = x^{2}, f^{'} (3) = ?$

6 Speed at the moment

A cart's position is x = t² + 3t. What is ẋ(2)?

$x = t^{2} + 3 t, x ˙ (2) = ?$

7 A thrown ball

A ball's height is h = 20t − 5t². What is d²h/dt²?

$h = 20 t - 5 t^{2}, \frac{d ^{2} h}{d t ^{2}} = ?$

8 Speed, two ways

Write the average speed over a stretch, and the speed at one moment, as a derivative. Then compare.

Your version

9 The cyclist who wobbles

A cyclist’s position along a lane is x = t³ − 6t² + 9t. Judge each sentence: true or false?

	$x ˙ > 0$	$x ˙ = 0$	$x ˙ < 0$	$\frac{d ^{2} x}{d t ^{2}} > 0$
$t = 0$	✓
$t = 1$		✓
$t = 2$			✓
$t = 3$		✓		✓
$t = 4$	✓			✓

A tick means the column’s claim holds at that moment, for x = t³ − 6t² + 9t. The speed is ẋ = 3t² − 12t + 9 and the acceleration is 6t − 12.

$x ˙ (1) = 0$
$x ˙ (2) > 0$
$\frac{d ^{2} x}{d t ^{2}} (3) > 0$
$x (3) - x (0) = Δ x = 0$
$\frac{x ( 1 ) - x ( 0 )}{1 - 0} = 4$
$x ˙ (4) < 0$

10 What the slope says

Five lines about the graph of f near x = 2. Pair each with what it says.

Tap a line, then its meaning.

11 Plug in the time

A sledge slides with x = 5t². This should give its speed at t = 3. One tile is wrong. Find it, then fix it.

Tap the tile that is wrong.

12 The cooling tea

A cup of tea cools from 20 to 12 degrees above the room in 4 minutes. This should give the average rate. One tile is wrong. Find it, then fix it.

Tap the tile that is wrong.

13 Where the road is flat

A road has height y = x³ − 3x. At which x is it flat?

$y = x^{3} - 3 x, \frac{d y}{d x} = 0 \Leftrightarrow x = ?$

14 Ten minutes of cooling

A kettle is 90 degrees at the start and 60 degrees after 10 minutes. What is the average rate of change of its temperature?

$T (0) = 90, T (10) = 60, \frac{Δ T}{Δ t} = ?$

15 The falling stone

A stone dropped from 45 metres has height h = 45 − 5t². What is ḣ(3)?

$h = 45 - 5 t^{2}, h ˙ (3) = ?$

16 Rate of a rate

Write in symbols: acceleration is the rate of change of speed, which is itself the rate of change of position. Then compare.

Your version

Tell your own

Tell a three-line story about something speeding up or slowing down, using delta x over delta t and x dot. Then read it aloud.

Kept on this device only.

How fast: derivatives

The alphabet

The story: The speedometer

Read it

Write it

Build it

Challenges

1 A walk there and back

2 Change in every dress

3 Average speed

4 The second time round

5 Slope at 3

6 Speed at the moment

7 A thrown ball

8 Speed, two ways

9 The cyclist who wobbles

10 What the slope says

11 Plug in the time

12 The cooling tea

13 Where the road is flat

14 Ten minutes of cooling

15 The falling stone

16 Rate of a rate

Tell your own

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