A. Rate of Change · Derivatives and Differentiation Rules

Derivatives and Differentiation Rules: Unit Review

A2.1 A2.2 A2.3 A3.1 A3.2 A3.3 A3.4 A3.5About 40 minutes

Learning goals

I can describe the derivative as a rate of change, a slope graph and a limit.
I can differentiate polynomial, rational and radical functions with the power, sum, product, quotient and chain rules.
I can use derivatives to find tangents, normals, higher derivatives and rates of change in context.

Words to know

derivative
The function f′ giving the instantaneous rate of change (tangent slope) of f at each x.

differentiation rules
The power, constant, constant multiple, sum, difference, product, quotient and chain rules.

rate of change
How fast one quantity changes compared with another, measured by the derivative.

Let's review!

In this unit you read the sign of the instantaneous rate of change to find where a function rises and falls, built the derivative function from tables of tangent slopes, and defined it exactly from first principles with a limit. You connected the graphs of f and f′. Then you learned the rules that make differentiation fast: the power rule (for natural, negative and rational exponents), the constant, constant multiple, sum and difference rules, and the product, quotient and chain rules. You used them for tangents and normals, rates with units, higher derivatives and real problems with polynomial, rational and radical models.

The next unit, Derivatives of Exponential and Sinusoidal Functions, adds eˣ, aˣ, ln x, sin x and cos x to your toolkit, and the same product, quotient and chain rules will apply to them. This review has a few questions from every lesson. Try each one, then press Show answer to check.

How to use this review

Do one or two sections at a time. Before differentiating, rewrite roots and reciprocals as powers and decide which rule comes first (look at the last operation). For each question you miss, write down why (missed an inner derivative? used f instead of f′? wrong sign in the quotient rule?) and go back to that lesson. Finish with the 10-question quiz. A scientific calculator is enough for every question.

How the unit fits together. Signs and tables of tangent slopes lead to the derivative function; first principles define it exactly; the rules make it fast; and the rules solve problems.

Every differentiation rule from the unit on one card. The power rule works for any rational exponent n.

Just for fun

Calculus students are said to be excellent at parties: they always know how fast things are changing, they can find the high point of any evening, and they never forget to bring the chain.

Increasing, Decreasing and the Sign of the Rate of Change

The rate of change of f is negative for x < 2 and positive for x > 2. What happens at x = 2?

For g(x) = −x² + 6x, the tangent slope at x = a is −2a + 6. Where is g increasing?

Does a zero rate of change always mean a maximum or minimum? Give an example.

The Derivative Function: Tables and Graphs of Tangent Slopes

Tangent slopes of f at x = 0, 1, 2, 3 are −3, −1, 1, 3. Find f′(x).

A cubic has a derivative of what degree?

Estimate f′(2) for f(x) = x³ with [f(2.001) − f(1.999)] ÷ 0.002.

The Derivative from First Principles

State the definition of f′(x).

Differentiate f(x) = 2x² − x from first principles.

Where is y = |x + 3| not differentiable, and why?

Connecting the Graphs of a Function and Its Derivative

f has a local max at x = −1 and a local min at x = 4. Where is f′ zero, and what is its sign between?

f′ is the line y = −2x + 6. Describe f.

Is the derivative of an even polynomial even or odd?

The Power Rule

Differentiate y = x⁹.

Find the tangent to y = x³ at x = −1.

Where on y = x⁴ is the slope 32?

The Constant, Constant Multiple, Sum and Difference Rules

Differentiate y = 3x⁴ − 5x³ + x − 8.

Differentiate y = (x − 2)(x + 5) by expanding.

Barrel A fills at 2.5 L/min and barrel B at 1.5 L/min. How fast is the total volume increasing?

Derivatives of Polynomials: Tangents, Normals and Points with a Given Slope

Find the normal to y = x² − 2x at x = 2.

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

At which point on y = x² + 5x is the tangent parallel to y = x − 4?

Polynomial Models: Rates of Change and Higher Derivatives

P(t) = 800 + 20t + t² (people, t in years). Find P′(10) with units.

For f(x) = x⁵, find f″(x) and f‴(x).

For C(x) = 500 + 8x − 0.002x², find the marginal cost at x = 1000.

The Power Rule for Negative and Rational Exponents

Differentiate y = 6∛x.

Differentiate y = 3/x² − √x.

Find the slope of y = 1/√x at x = 4.

The Product Rule

Differentiate y = (x² + 2)(3x − 1).

If f(2) = 1, f′(2) = 5, g(2) = −3, g′(2) = 2, find (fg)′(2).

Differentiate y = x√x two ways: product rule and power rule.

The Chain Rule

Differentiate y = (2x² − 1)⁴.

Differentiate y = √(9 − x²) and find the slope at x = 0.

y = u³ and u = 1 − 2x. Find dy/dx at x = 0.

The Quotient Rule and Rational Functions

Differentiate y = (x + 3)/(x − 1).

Differentiate y = 2x/(x² + 1) and find its horizontal tangents.

Rewrite (x² − 4)/(x + 5) as a product.

Combining the Rules: Radical and Rational Functions

Which rule first for y = x³√(x + 1)? Differentiate.

Differentiate y = [(x − 2)/(x + 2)]².

Find y″ for y = 1/(2x − 1).

Solving Rate-of-Change Problems with the Derivative Rules

V(t) = 300(1 − t/30)² litres. How fast is the tank draining at t = 15 min?

A plant's height is h(t) = 40t/(t + 5) cm, t in months. Find h′(5) with units.

A question asks "when is the rate 3?" What do you do?

Check your understanding

Quiz

Where the tangent slopes of a smooth function are positive, the function is:

Which limit defines f′(x)?

The zeros of f′ correspond on the graph of f to:

What is d/dx (4x⁵ − 2x + 7)?

What is the derivative of y = 2√x?

What is the derivative of y = x²(x + 1)?

What is the derivative of y = (3x + 2)⁵?

What is the derivative of y = x/(x + 4)?

The tangent to y = x² − 3x at x = 2 is:

10.

For f(x) = x⁴, what is f″(1)?

For parents and teachers

Use the review to find what to revisit:

Confusing a function's value with its rate, or calling every zero slope a turning point → Lesson 1 (Increasing, Decreasing and the Sign of the Rate of Change).
Plotting y-values instead of slopes when building a slope graph → Lesson 2 (The Derivative Function).
Algebra or limit errors in first-principles questions → Lesson 3 (The Derivative from First Principles).
Sketching f′ under the x-intercepts of f instead of under its turning points → Lesson 4 (Connecting the Graphs).
Power-rule slips (exponent not lowered, signs with odd powers) → Lesson 5 (The Power Rule).
Keeping constants, or multiplying derivatives of factors → Lesson 6 (Constant, Sum and Difference Rules).
Using f′(a) as the y-coordinate, or −m instead of −1/m for normals → Lesson 7 (Tangents, Normals and Given Slopes).
Missing units, or mixing average and instantaneous rates → Lesson 8 (Rates of Change and Higher Derivatives).
Rewriting roots and reciprocals as powers incorrectly → Lesson 9 (Negative and Rational Exponents).
Writing (fg)′ = f′g′ → Lesson 10 (The Product Rule).
Forgetting the inner derivative → Lesson 11 (The Chain Rule).
Sign errors in the quotient rule numerator → Lesson 12 (The Quotient Rule and Rational Functions).
Not knowing which rule to use first → Lesson 13 (Combining the Rules).
Setting up word problems, or answering without units → Lesson 14 (Solving Rate-of-Change Problems).

Looking ahead: Derivatives of Exponential and Sinusoidal Functions extends these rules to eˣ, ln x, sin x and cos x; Curve Sketching: Connecting f, f′ and f″ uses f′ and f″ to draw graphs; Applications of Derivatives covers motion, related rates and optimization.