Rules for Differentiation Revision Notes for Grade 12 NSC Matric Mathematics

Rules for Differentiation

Introduction to differentiation rules

Calculating derivatives using first principles involves lengthy calculations and is prone to errors. Differentiation rules provide shortcuts that make finding derivatives much simpler and more reliable. These rules are derived from first principles but allow us to differentiate functions quickly and accurately.

When we use differentiation rules, we can find the derivative of a function without going through the limit definition every time. This makes calculus much more practical for solving real-world problems.

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The beauty of differentiation rules is that they transform what could be hours of complex limit calculations into simple, systematic procedures that can be applied consistently across different types of functions.

The five essential differentiation rules

1. The power rule

The most fundamental rule in differentiation is the power rule:

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

where $n \in \mathbb{R}$ and $n \neq 0$

This means when differentiating $x$ raised to any power:

Multiply by the current power
Reduce the power by 1

Example: $\frac{d}{dx}[x^5] = 5x^4$

2. The constant rule

The derivative of any constant is zero:

$\frac{d}{dx}[k] = 0$

This makes sense because constants don't change, so their rate of change is zero.

Example: $\frac{d}{dx}[60] = 0$

3. The constant multiple rule

When a constant multiplies a function, the constant stays and multiplies the derivative:

$\frac{d}{dx}[k \cdot f(x)] = k \cdot \frac{d}{dx}[f(x)]$

The constant "comes along for the ride" when differentiating.

Example: $\frac{d}{dx}[3x^2] = 3 \cdot \frac{d}{dx}[x^2] = 3 \cdot 2x = 6x$

4. The sum rule

The derivative of a sum equals the sum of the derivatives:

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

You can differentiate each term separately and add the results.

Example: $\frac{d}{dx}[x^3 + 2x] = 3x^2 + 2$

5. The difference rule

The derivative of a difference equals the difference of the derivatives:

$\frac{d}{dx}[f(x) - g(x)] = \frac{d}{dx}[f(x)] - \frac{d}{dx}[g(x)]$

Similar to the sum rule, differentiate each term separately and subtract.

Example: $\frac{d}{dx}[x^4 - 5x] = 4x^3 - 5$

When to use rules versus first principles

Understanding when to use differentiation rules versus first principles is crucial for exam success:

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Use differentiation rules when:

The question doesn't specify a method
You need to find derivatives quickly
Working with standard polynomial, rational, or simple functions

Use first principles when:

The question explicitly asks for first principles
You need to prove or derive a differentiation rule
The question asks you to "use the definition of a derivative"

Worked examples

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Worked Example 1: Basic applications

Find the derivatives of the following functions:

1. $y = 3x^5$

Apply the power rule and constant multiple rule
$\frac{dy}{dx} = 3 \times 5x^4 = 15x^4$

2. $p = \frac{1}{4}q^2$

Rewrite as $p = 0.25q^2$
$\frac{dp}{dq} = 0.25 \times 2q = 0.5q = \frac{1}{2}q$

3. $f(x) = 60$

This is a constant
$f'(x) = 0$

4. $y = 12x^3 + 7x$

Use the sum rule
$\frac{dy}{dx} = 12 \times 3x^2 + 7 \times 1 = 36x^2 + 7$

5. $m = \frac{2}{3}n^4 - 1$

Use the difference rule and constant rule
$\frac{dm}{dn} = \frac{2}{3} \times 4n^3 - 0 = \frac{8}{3}n^3$

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Worked Example 2: Functions requiring expansion

Sometimes you need to expand expressions before applying differentiation rules.

Find $g'(t)$ if $g(t) = 4(t + 1)^2(t - 3)$

Step 1: Expand the expression first

$g(t) = 4(t + 1)^2(t - 3)$
$= 4(t^2 + 2t + 1)(t - 3)$
$= 4(t^3 + 2t^2 + t - 3t^2 - 6t - 3)$
$= 4(t^3 - t^2 - 5t - 3)$
$= 4t^3 - 4t^2 - 20t - 12$

Step 2: Apply the differentiation rules

$g'(t) = 4(3t^2) - 4(2t) - 20(1) - 0$
$= 12t^2 - 8t - 20$

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Worked Example 3: Fractional and negative exponents

Find $k'(t)$ if $k(t) = \frac{(t + 2)^3}{\sqrt{t}}$

Step 1: Expand and simplify

$k(t) = \frac{(t + 2)^3}{t^{1/2}}$
$= \frac{(t + 2)(t^2 + 4t + 4)}{t^{1/2}}$
$= \frac{t^3 + 6t^2 + 12t + 8}{t^{1/2}}$
$= t^{5/2} + 6t^{3/2} + 12t^{1/2} + 8t^{-1/2}$

Step 2: Apply the power rule to each term

$k'(t) = \frac{5}{2}t^{3/2} + 6 \cdot \frac{3}{2}t^{1/2} + 12 \cdot \frac{1}{2}t^{-1/2} + 8 \cdot \left(-\frac{1}{2}\right)t^{-3/2}$
$= \frac{5}{2}t^{3/2} + 9t^{1/2} + 6t^{-1/2} - 4t^{-3/2}$

Final answer with positive exponents: $k'(t) = \frac{5}{2}t^{3/2} + 9t^{1/2} + \frac{6}{t^{1/2}} - \frac{4}{t^{3/2}}$

Important exam tips

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Essential exam strategies:

Always write final answers with positive exponents unless specifically asked otherwise
Show your working clearly - examiners award method marks
Check if expansion is needed before applying rules
Be careful with negative signs when using the difference rule
Remember that constants disappear when differentiating

Remember!

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Key Points to Remember:

The power rule $\frac{d}{dx}[x^n] = nx^{n-1}$ is your most important tool
Constants have zero derivatives - they completely disappear
Constant multiples stay - multiply the derivative by the constant
Sums and differences split - differentiate each term separately
Expand complex expressions first before applying the rules
Always write final answers with positive exponents

Rules for Differentiation (Grade 12 NSC Matric Mathematics): Revision Notes