Graphing the Derivative Function

Given the graph of a function $f$ , we can sketch the graph of its derivative f' by reading off tangent-line slopes at each point. This section explains the strategy and demonstrates it with two detailed examples.

Feature of $f$	Corresponding feature of f'
Horizontal tangent (slope $= 0$ )	f' = 0 (zero of f')
$f$ increasing (slope $> 0$ )	f' > 0
$f$ decreasing (slope $< 0$ )	f' < 0
Slope becoming steeper	$
Slope becoming shallower	$

The Key Idea

The original function $f (x)$ provides the height of the curve $y = f (x)$ at the value $x$ . If we measure the slope of the tangent line to the curve $y = f (x)$ at a variable point $(x, f (x))$ , we obtain a new function called the derivative, denoted f'. Hence:

$f (x)$ = the height of the curve at the point $x$ .
f'(x) = the slope of the curve at that same point $x$ .

If the graph of $f$ is given, we can draw a sketch of the graph of f' by estimating the slope of the tangent to the graph of $f$ at each $x$ value. We then plot the points (x, f'(x)) in the xy'-plane and connect them by a smooth curve whenever possible. This curve represents the graph of f'.

Formally:

f (x) = height of the curve y = f (x) at the point x

f'(x) = \text{slope of the curve } y = f(x) \text{ at the point } x. Therefore, if the graph of

f

is given, the set of points

(x, slope of tangent to the graph of f at x)

forms the graph of f'.

When seeking to graph the derivative, it is often easiest to first identify the places where the tangent lines are horizontal and thus their slopes are zero. Those are the places where the sign of the derivative may change.

Examples

The graph of a function $f$ is shown below. Give a rough sketch of the graph of f'.

Graph of function f showing a curve with a local maximum near x=0 and a local minimum near x=2

Solution

The tangent to the curve is horizontal at

x = 0

and

x = 2

. That is, f'(0) = f'(2) = 0.

When $x < 0$ , the tangent line makes an acute angle $θ$ with the positive $x$ -axis. Thus f'(x) = \tan\theta > 0.
When $0 < x < 2$ , the tangent line makes an obtuse angle $θ$ with the positive $x$ -axis. Thus f'(x) = \tan\theta < 0.
When $2 < x$ , the tangent line makes an acute angle $θ$ with the positive $x$ -axis. Thus f'(x) = \tan\theta > 0.

Specifically, we can read off several slopes:

At $x = - 1$ : slope $= \frac{rise}{run} = \frac{3}{1}$ , so f'(-1) = 3.
At $x = 1$ : slope $= \frac{- 1}{1}$ , so f'(1) = -1.
At $x = 3$ : slope $= \frac{- 3}{- 1} = 3$ , so f'(3) = 3.

Graph of f with tangent lines drawn at x equals negative 1, 0, 1, 2, and 3 to illustrate slope readings

Connecting the points $(-1, 3),\; (0, 0),\; (1, -1),\; (2, 0),\; (3, 3),$ we sketch the graph of f':

Rough sketch of the derivative f prime, positive for x less than 0, zero at x=0, negative between 0 and 2, zero at x=2, positive for x greater than 2

The graph of a function $f$ is shown below. Give a rough sketch of the graph of f'. (The sketch does not need to be exact; just show its main features.)

Graph of a more complex function f with local extrema near x equals negative 1, negative 0.2, and 1

Solution

The graphs of

f

and its derivative are shown below:

Graph of f and its derivative f prime plotted together

The slope of the tangent line is zero at

x = - 1

x = - 0.2

, and

x = 1

, giving three points on the graph of f':

(- 1, 0)

(- 0.2, 0)

, and $(1, 0)$.

For $x < - 1$ : the tangent makes an acute angle with the positive $x$ -axis, so f'(x) > 0. As $x$ becomes large and negative, f' is large positive; as $x \to - 1^{-}$ , f' decreases to 0.
For $- 1 < x < - 0.2$ : the tangent makes an obtuse angle, so f'(x) < 0. The steepest slope in this interval occurs at point $B$ , so f' has a minimum there.
For $- 0.2 < x < 1$ : the tangent makes an acute angle, so f'(x) > 0. The slope first increases, reaching a local maximum at point $C$ , then decreases until it reaches zero at $x = 1$ .
For $x > 1$ : the slope is positive and grows larger as $x$ increases.

Frequently Asked Questions

How do I know where to place the zeros of f'?

The zeros of f' occur wherever

f

has a horizontal tangent line. These are typically the local maxima and local minima of

f

, as well as any other point where the curve momentarily levels off.

How do I determine the sign of f' between zeros?

Between consecutive zeros of f', pick any one convenient

x

value, read the approximate tangent slope from the graph of

f

, and note whether it is positive or negative. That sign holds throughout the entire interval (unless f' crosses zero again in between).

Does the graph of f' have to be smooth?

Not necessarily. If the graph of

f

has a corner (a sharp angle), then f' will have a jump discontinuity at that point (or will be undefined there). The graph of f' is smooth only where the graph of

f

has a smoothly varying slope.

The Key Idea

Examples

Frequently Asked Questions

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