A. Rate of Change · Derivatives and Differentiation Rules

The Constant, Constant Multiple, Sum and Difference Rules

A3.2About 65 minutes

Learning goals

I can state and use the constant, constant multiple, sum and difference rules.
I can verify these rules graphically and numerically.
I can differentiate any polynomial term by term.
I can interpret the sum rule in a real context, such as two containers filling.

Words to know

constant rule
The derivative of a constant function is 0.

constant multiple rule
d/dx [k f(x)] = k f′(x).

sum rule
d/dx [f(x) + g(x)] = f′(x) + g′(x).

difference rule
d/dx [f(x) − g(x)] = f′(x) − g′(x).

term by term
Differentiating each term of a sum or difference separately and combining the results.

Building any polynomial from powers

Every polynomial is made from powers of x using three moves: multiplying by a constant (3x²), adding (x³ + 3x²) and subtracting (x³ − 3x² − 5). If we know how each move affects the derivative, the power rule lets us differentiate any polynomial in one line.

In this lesson you will verify four rules graphically and numerically. The proofs are short and are included for interest; the curriculum asks you to understand and use them, not to memorize the proofs.

The four rules

For differentiable functions f and g and any constant k:

Constant rule: d/dx (k) = 0.
Constant multiple rule: d/dx [k f(x)] = k f′(x).
Sum rule: d/dx [f(x) + g(x)] = f′(x) + g′(x).
Difference rule: d/dx [f(x) − g(x)] = f′(x) − g′(x).

Seeing the rules on graphs

Constant rule. y = 5 is a horizontal line. Its slope is 0 everywhere.
Constant multiple rule. Multiplying a function by k is a vertical stretch by factor k (Grade 11 transformations). Every rise is multiplied by k while every run is unchanged, so every slope is multiplied by k.
Sum rule. If two quantities are both changing, the total changes at the sum of the two rates. In the graph of y = f(x) + g(x), each small rise is the rise of f plus the rise of g over the same run.

Stretching y = x² vertically by a factor of 3 gives y = 3x². Every rise triples while the run stays the same, so every tangent slope triples: at x = 1 the slopes are 2 and 6. That is the constant multiple rule.

x	−2	−1	1	2
slope of x²	−4	−2	2	4
slope of 3x²	−12	−6	6	12
slope of x³	12	3	3	12
slope of x³ + 3x²	0	−3	9	24

Numerical check (slopes from centred estimates, rounded). Each slope of 3x² is 3 times the slope of x², and each slope of x³ + 3x² is the sum of the slopes of x³ and 3x².

Why the sum rule works (for interest)

Let s(x) = f(x) + g(x). Then

[s(x + h) − s(x)] ÷ h = [f(x + h) − f(x)] ÷ h + [g(x + h) − g(x)] ÷ h.

As h → 0, the first quotient approaches f′(x) and the second approaches g′(x), and the limit of a sum is the sum of the limits. So s′(x) = f′(x) + g′(x). The constant multiple rule works the same way: [kf(x + h) − kf(x)] ÷ h = k × [f(x + h) − f(x)] ÷ h → kf′(x).

Worked example

Example 1: Differentiating a polynomial term by term

Differentiate f(x) = 4x³ − 5x² + 7x − 9.

Difference and sum rules: differentiate each term separately.
4x³ → 4(3x²) = 12x² (constant multiple and power rules).
−5x² → −5(2x) = −10x.
7x → 7(1) = 7.
−9 → 0 (constant rule).

f′(x) = 12x² − 10x + 7

Worked example

Example 2: Two rain barrels

Two rain barrels collect water during a spring storm. Barrel A holds V_A(t) = 2t + 0.1t² litres and barrel B holds V_B(t) = 3t litres after t minutes, 0 ≤ t ≤ 20. (a) Find the rate at which each barrel is filling at t = 10. (b) Use the sum rule to find the rate at which the total volume is increasing at t = 10, and check by differentiating the total directly.

(a) V_A′(t) = 2 + 0.2t, so V_A′(10) = 4 L/min. V_B′(t) = 3, so V_B′(10) = 3 L/min.
(b) Sum rule: the total rate is 4 + 3 = 7 L/min.
Check: V(t) = V_A(t) + V_B(t) = 5t + 0.1t², so V′(t) = 5 + 0.2t and V′(10) = 5 + 2 = 7 L/min. ✓
Meaning: V_A′ and V_B′ are each barrel's filling rate; the sum rule says the rates simply add.

(a) 4 L/min and 3 L/min; (b) 7 L/min.

Two rain barrels fill at the same time. The total volume is the sum of the two volumes, and the total rate of filling is the sum of the two rates. This is the sum rule in a real setting.

Worked example

Example 3: Expand first

Differentiate y = (2x − 1)(x + 3).

There is no rule yet for a product, so expand: y = 2x² + 6x − x − 3 = 2x² + 5x − 3.
dy/dx = 4x + 5.
Warning: multiplying the derivatives of the factors gives (2)(1) = 2, which is wrong. You will learn the correct product rule in the lesson The Product Rule.

dy/dx = 4x + 5

Worked example

Example 4: Leibniz notation and other letters

Find ds/dt if s = −4.9t² + 15t + 2, and evaluate it at t = 1.

Differentiate with respect to t: ds/dt = −4.9(2t) + 15 + 0 = −9.8t + 15.
At t = 1: −9.8 + 15 = 5.2.

ds/dt = −9.8t + 15; at t = 1 it is 5.2. (If s is the height in metres of a ball thrown upward, this is its velocity in m/s. You will study motion fully in Applications of Derivatives.)

Just for fun

Ask a calculus student to differentiate their age in years and they will tell you, correctly, that the rate of change is 1 year per year. Everyone gets older at exactly the same rate; it only feels faster in exam season.

Hands-on activity

Stretch and add with a ruler

You need: graph paper, a pencil, a ruler and a calculator.

Draw y = x² and y = 2x² on the same axes for 0 ≤ x ≤ 2.5 (plot x = 0, 0.5, 1, 1.5, 2, 2.5).
At x = 1.5, lay your ruler along the tangent to each curve and estimate each slope by counting squares.
Is the second slope about twice the first? Compare with 2(1.5) = 3 and 4(1.5) = 6.
Now draw y = x² + x on the same axes. Estimate its slope at x = 1.5.
Check: the sum rule predicts 3 + 1 = 4.

What to look for: slopes near 3, 6 and 4. Hand-drawn tangents are rarely perfect, so being within about 0.5 is good. The point of the activity is the pattern: stretching doubles slopes and adding functions adds slopes.

Your turn

Differentiate y = 6x⁴.

Differentiate f(x) = x³ − 8x + 2.

Differentiate g(x) = −2x⁵ + 3x² − x.

Differentiate y = 0.5x² − 7.

Find f′(2) for f(x) = 3x³ − 4x² + 1.

Differentiate y = (x + 4)² by expanding first.

If h(x) = f(x) − g(x), f′(3) = 7 and g′(3) = −2, find h′(3).

More practice: warm-up

d/dx (12) = ?

d/dx (5x) = ?

d/dx (−3x²) = ?

d/dx (x² + x) = ?

If f′(x) = 4x, what is the derivative of 3f(x)?

More practice: on your own

Differentiate y = (x² − 3)(x² + 3).

Differentiate f(x) = x(x − 2)(x + 2).

Find the equation of the tangent to y = 2x³ − 3x² − 12x + 5 at x = −1.

Find the points on y = x³ − 3x² + 4 where the tangent is horizontal.

In the rain-barrel example, at what time is barrel A filling at the same rate as barrel B?

Challenge

Find a and b if the parabola y = ax² + bx has tangent slope 7 at x = 1 and tangent slope −1 at x = −1.

Find the point on y = x³ − 6x² + 5 where the tangent slope is smallest, and state that slope.

Find the equations of both tangents to y = x² that pass through the point (1, −3), which is not on the curve.

Check your understanding

Quiz

What is the derivative of f(x) = 5x³ − 2x + 8?

Graphically, the constant multiple rule says that a vertical stretch by a factor of k:

If f′(2) = 5 and g′(2) = 3, what is the derivative of f(x) + g(x) at x = 2?

Barrel A fills at 4 L/min and barrel B at 2.5 L/min at the same moment. How fast is the total volume increasing?

What is dy/dx for y = (x − 1)(x + 1)?

Where is the tangent to y = x² − 6x + 1 horizontal?

For parents and teachers

Big idea (A3.2): the derivative of a sum or difference is the sum or difference of the derivatives, a constant multiple stays in front, and constants have derivative 0. Together with the power rule, this lets your teen differentiate any polynomial term by term.

How to help: Write a polynomial such as 3x⁴ − 2x² + x − 6 and ask your teen to differentiate it aloud term by term. Then ask what the derivative means: in the barrel example, it is litres per minute.

Common difficulties:

Keeping a constant term after differentiating (the derivative of −9 is 0, not −9).
Forgetting that the derivative of 7x is 7, not 7x or 0.
Multiplying derivatives of factors instead of expanding first (the product rule is taught in the lesson The Product Rule).
Sign errors with negative coefficients.