The derivative function

Warm-up

Three quick questions from earlier lessons.

Discover

The violet curve is $y = x^3 - 3x$ . The blue line is its tangent at $x = a$ . Slide $a$ from $-2$ to $2$ and watch the slope.

The cubic y = x³ − 3x and its tangent at x = a. The tangent is horizontal at x = −1 and x = 1, slopes down between them, and slopes up outside.

Between $x = -1$ and $x = 1$ , the graph falls. What is the sign of the slope there?

Negative
Positive

Key idea

Computing $f'(a)$ at every $a$ gives a new function.

The derivative function is

f'(x) = \displaystyle\lim_{h \to 0} \dfrac{f(x + h) - f(x)}{h}

. Where

f

rises,

f'(x) > 0

; where

f

falls,

f'(x) < 0

; where the tangent is horizontal,

f'(x) = 0

.

Worked example

$f(x) = x^2 - 4x$ . Find $f'(x)$ .

f(x + h) - f(x) = 2xh + h^2 - 4h

Expand (x + h)² − 4(x + h) and subtract x² − 4x.

\dfrac{f(x + h) - f(x)}{h} = 2x + h - 4

Divide by h ≠ 0.

f'(x) = 2x - 4

Let h → 0. Check: f' = 0 at x = 2, the vertex.

Your turn, guided

Your turn: $f(x) = x^2 + 6x$ .

\dfrac{f(x + h) - f(x)}{h} = 2x + h + 6

The same steps.

f'(x) = 2x + 6

Let h → 0.

Practice ★

Here is the graph of $y = f(x)$ . Tap the point of the graph where $f'(x) = 0$ .
Here is the graph of $y = f(x)$ . Tap the point of the graph where $f'(x) = 0$ .

$f(x) = (x - 3)^2 + 3$ , a parabola with its vertex at $(3, 3)$ . Tick all the true statements.
1. $f'(3) = 0$
2. $f'(x) < 0$ for $x < 3$
3. $f'(x) > 0$ for $x > 3$
4. $f'(3) = 3$
5. $f'(x) > 0$ for every $x$ , because $f(x) > 0$

Watch out!

Practice ★★

$f(x) = 3x^2 + 8x$ . Use $f'(x) = \displaystyle\lim_{h \to 0} \dfrac{f(x + h) - f(x)}{h}$ to find $f'(x)$ .
$f(x) = 3x^2 + 5x$ . Use $f'(x) = \displaystyle\lim_{h \to 0} \dfrac{f(x + h) - f(x)}{h}$ to find $f'(x)$ .
$f(x) = 5x^2 + 2x$ . Use $f'(x) = \displaystyle\lim_{h \to 0} \dfrac{f(x + h) - f(x)}{h}$ to find $f'(x)$ .

Exit ticket

Three questions to check you have got it.

$f(x) = 3x^2 + 8x$ . Use $f'(x) = \displaystyle\lim_{h \to 0} \dfrac{f(x + h) - f(x)}{h}$ to find $f'(x)$ .
$f(x) = 3x^2 + 5x$ . Use $f'(x) = \displaystyle\lim_{h \to 0} \dfrac{f(x + h) - f(x)}{h}$ to find $f'(x)$ .
$f(x) = 5x^2 + 2x$ . Use $f'(x) = \displaystyle\lim_{h \to 0} \dfrac{f(x + h) - f(x)}{h}$ to find $f'(x)$ .

Summary card

The derivative function

$f'(x)$ is the slope of $f$ at each $x$ .
$f$ rises: $f' > 0$ . $f$ falls: $f' < 0$ . Horizontal tangent: $f' = 0$ .

Watch out: $f' = 0$ (flat) is not the same as $f = 0$ (on the axis).

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