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Grade 12 Calculus · Lesson 2.2 + 2.3120 min

Differentiation Rules · Build the Toolbox

Keep the meaning of derivative from Lesson 2.1, but replace repeated limit work with efficient rules. Read the structure, rewrite when useful, choose the correct rule, then differentiate and interpret.

Build fluency with the basic differentiation rules, trigonometric derivatives, Product and Quotient Rules, higher-order derivatives, and motion applications.

Core rulesRewrite firstTrig derivativesProduct RuleQuotient RuleHigher orderMotion

Key takeaway

Mr. Azzam’s rule-choice routine

1 · Inspect

Identify the structure before touching the derivative.

2 · Rewrite

Convert simple radicals and fractions to powers when useful.

3 · Choose

Use the simplest valid differentiation rule.

4 · Differentiate + check

Simplify, then check signs, exponents, notation, and meaning.

Build the differentiation toolbox

In Lesson 2.1 you built derivatives from limits. Now we use rules that produce the same derivative efficiently.

ExplanationConstant Rule

\frac{d}{dx}[c]=0

ExplanationPower Rule

\frac{d}{dx}[x^n]=nx^{n-1}

ExplanationConstant Multiple Rule

\frac{d}{dx}[cf(x)]=cf'(x)

ExplanationSum & Difference Rules

\frac{d}{dx}[f(x)\pm g(x)]=f'(x)\pm g'(x)

Worked example · Example · Differentiate term by term

f(x)=5x^4-3x^2+7x-9

Predict the next idea, or open the worked steps whenever you need them.

Predict before you reveal

For

f(x)=5x^4-3x^2+7x-9

, what does the constant term contribute to

f'(x)

-9

0

-9x

Your turn

Differentiate

f(x)=7x^5

7x^4

35x^4

35x^5

Mr. Azzam’s routine: rewrite before differentiating

Before choosing a rule, inspect the expression. Fractions and radicals can often be rewritten as powers so the Power Rule becomes immediate.

Explanation

\frac1{x^m}=x^{-m}

Explanation

\sqrt{x}=x^{1/2}

Explanation

\sqrt[n]{x^m}=x^{m/n}

Worked example · Example · Rewrite first

f(x)=\frac{4}{x^3}+6\sqrt{x}

Predict the next idea, or open the worked steps whenever you need them.

Predict before you reveal

For

4/x^3+6\sqrt{x}

, where is the real derivative formula valid?

x\ge0

All real

x

x>0

Common mistakeDecision check

A visible fraction does not automatically mean Quotient Rule. If the denominator is only a power of x, rewrite first.

Your turn

Which is the simplest rewrite-first approach to differentiating

1/x^3

Differentiate numerator and denominator separatelyUse the Quotient Rule without simplifyingRewrite it as

x^{-3}

Trigonometric derivatives

\frac{d}{dx}(\sin x)=\cos x

\frac{d}{dx}(\cos x)=-\sin x

\frac{d}{dx}(\tan x)=\sec^2x

\frac{d}{dx}(\cot x)=-\csc^2x

\frac{d}{dx}(\sec x)=\sec x\tan x

\frac{d}{dx}(\csc x)=-\csc x\cot x

Use radians for these derivative formulas. For now the angle is simply $x$ . Expressions such as $\sin(3x^2)$ belong to Lesson 2.4: The Chain Rule.

Your turn

Find

\frac{d}{dx}(4\sin x-3\cos x)

, with angles in radians.

4\cos x+3\sin x

4\cos x-3\sin x

4\sin x+3\cos x

The Product Rule

\frac{d}{dx}[f(x)g(x)]=f'(x)g(x)+f(x)g'(x)

Common mistake. The derivative of a product is not generally the product of the derivatives.

Worked example · Example · Product Rule

y=x^2\sin x

Predict the next idea, or open the worked steps whenever you need them.

Predict before you reveal

For

y=x^2\sin x

, which pair identifies the two factors?

x^2

and

\sin x

2x

and

\cos x

x^2

and

\cos x

Key takeawayStructure first

Label the two factors before differentiating. Product Rule keeps both derivative contributions; differentiating only one factor is incomplete.

Your turn

Which setup is correct for

y=x^2\sin x

y'=2x\cos x

y'=2x\sin x+x^2\cos x

y'=x^2\cos x

The Quotient Rule

\frac{d}{dx}\left[\frac{f(x)}{g(x)}\right]=\frac{g(x)f'(x)-f(x)g'(x)}{[g(x)]^2}

Worked example · Example · Quotient Rule

g(x)=\frac{x-1}{x+1}

Predict the next idea, or open the worked steps whenever you need them.

Predict before you reveal

For

g(x)=(x-1)/(x+1)

, what denominator must the unsimplified Quotient Rule derivative have?

x+1

(x+1)^2

(x-1)^2

Common mistake. Keep the order in the numerator. Reversing the two products changes the sign of the derivative.

Common mistakeKeep the order

Denominator × derivative of numerator − numerator × derivative of denominator, all over denominator squared.

Your turn

For

y=\frac{x-1}{x+1}

, which numerator appears after the Quotient Rule?

(x-1)-(x+1)

1

(x+1)-(x-1)

Higher-order derivatives

A derivative is itself a function, so it can be differentiated again.

f(x)\longrightarrow f'(x)\longrightarrow f''(x)\longrightarrow f'''(x)\longrightarrow f^{(4)}(x)\longrightarrow\cdots

ExplanationHigher-order notation

Prime notation is normally used for the first three derivatives. Starting with the fourth derivative, write the order in parentheses rather than adding more prime marks.

f^{(4)}(x)\quad\text{not }f''''(x)

f^{(n)}(x)\quad\text{for the }n\text{th derivative}

Worked example · Position → velocity → acceleration

s(t)=-0.81t^2+2

Predict the next idea, or open the worked steps whenever you need them.

Predict before you reveal

For

s(t)=-0.81t^2+2

, what happens after differentiating the velocity once more?

Keep

-1.62t

Return to

-0.81t^2+2

Obtain the constant acceleration

-1.62

Your turn

s(t)

is position, what does

s''(t)

represent?

AccelerationVelocityPosition

Rates of change & motion

The derivative measures rate of change. For straight-line motion, the derivative of position is velocity; the derivative of velocity is acceleration.

v(t)=s'(t),\qquad a(t)=v'(t)=s''(t),\qquad \text{speed}=|v(t)|

Common mistake. Velocity can be negative, zero, or positive. Speed is the absolute value of velocity and cannot be negative.

Your turn

At one instant,

v=-6

m/s and

a=2

m/s

^2

. What is the speed?

-6

m/s

2

m/s

6

m/s

-12

m/s

Exam strategy & common mistakes

Key takeaway1 · Look at the structure before differentiating.

Can you rewrite it as powers? Is it a sum, product, quotient, or trig function?

Key takeaway2 · Choose the rule, then differentiate.

Do not use Product or Quotient Rule just because you see multiplication or a fraction if algebra can simplify the expression first.

Key takeaway3 · Simplify the derivative.

Combine like terms and, when appropriate, remove negative exponents from the final form.

Common mistake. Do not use the Chain Rule yet. Composite-function differentiation belongs to Lesson 2.4.

Build the differentiation toolbox

Worked example · Example · Differentiate term by term

1. Apply each rule

2. Simplify the derivative

Your turn

Your turn · student questions

Mr. Azzam’s routine: rewrite before differentiating

Worked example · Example · Rewrite first

1. Rewrite as powers

2. Apply the Power Rule

3. Write positive exponents

Your turn

Your turn · student questions

Trigonometric derivatives

Your turn

Your turn · student questions

The Product Rule

Worked example · Example · Product Rule

1. Differentiate one factor at a time

2. Combine the two contributions

Your turn

Your turn · student questions

The Quotient Rule

Worked example · Example · Quotient Rule

1. Keep the Quotient Rule order

2. Simplify the numerator

Your turn

Your turn · student questions

Higher-order derivatives

Worked example · Position → velocity → acceleration

1. Differentiate position for velocity

2. Differentiate velocity for acceleration

Your turn

Your turn · student questions

Rates of change & motion

Your turn

Your turn · student questions

Exam strategy & common mistakes