Derivative Notation and Basic Rules Revision Notes for HSC SSCE Mathematics Advanced

Derivative Notation and Basic Rules

Understanding how to differentiate functions efficiently is essential in calculus. This note introduces the power rule for differentiation and explores different ways to express derivatives using various notations. You'll also learn fundamental rules that allow you to differentiate more complex functions by breaking them down term by term.

The power rule

The power rule is one of the most important rules in differentiation. It provides a straightforward method for finding the derivative of any function where $x$ is raised to a power. This rule eliminates the need to use first principles or graphical approximations for each case.

The power rule states:

If $f(x) = x^n$ , then $f'(x) = nx^{n-1}$

where:

$f'(x)$ is the derivative of $f(x)$
$n$ is any real number

Alternatively, using different notation:

If $y = x^n$ , then $\frac{dy}{dx} = nx^{n-1}$

where:

$\frac{dy}{dx}$ is the derivative of $y$ with respect to $x$
$n$ is any real number

To apply the power rule, bring down the exponent as a multiplier in front of the term, then reduce the exponent by one. For example, if you're differentiating $x^5$ , multiply by $5$ to get $5x$ , then subtract $1$ from the power to get $5x^4$ .

infoNote

This rule works for all real powers including positive integers, negative numbers, and fractions. The key is ensuring your function is written in the form $x^n$ before applying the rule.

Worked example: Using the power rule

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Worked Example: Applying the Power Rule

Let's work through differentiating several functions using the power rule.

Part a: Differentiate $y = x^4$

Write the function: $y = x^4$

Apply the power rule $\frac{dy}{dx} = nx^{n-1}$ : $\frac{dy}{dx} = 4x^{4-1}$

Simplify the exponent: $\frac{dy}{dx} = 4x^3$

The derivative of $y = x^4$ is $\frac{dy}{dx} = 4x^3$ .

Part b: Differentiate $y = \frac{1}{x^2}$

First, rewrite the function using the negative index rule. Recall that $\frac{1}{x^n} = x^{-n}$ .

Write the function: $y = \frac{1}{x^2}$

Use the negative index rule: $y = x^{-2}$

Apply the power rule $\frac{dy}{dx} = nx^{n-1}$ : $\frac{dy}{dx} = -2x^{-2-1}$

Simplify the exponent: $\frac{dy}{dx} = -2x^{-3}$

The derivative can be left in index form as $\frac{dy}{dx} = -2x^{-3}$ , or rewritten with a positive index as $\frac{dy}{dx} = \frac{-2}{x^3}$ .

Part c: Differentiate $y = \sqrt{x}$

First, convert the square root to index form. Recall that $\sqrt{x} = x^{\frac{1}{2}}$ .

Write the function: $y = \sqrt{x}$

Use the rule $\sqrt{x} = x^{\frac{1}{2}}$ : $y = x^{\frac{1}{2}}$

Apply the power rule $\frac{dy}{dx} = nx^{n-1}$ : $\frac{dy}{dx} = \frac{1}{2}x^{\frac{1}{2}-1}$

Simplify the exponent: $\frac{dy}{dx} = \frac{1}{2}x^{-\frac{1}{2}}$

The derivative can be written as $\frac{dy}{dx} = \frac{1}{2}x^{-\frac{1}{2}}$ or in surd form as $\frac{dy}{dx} = \frac{1}{2\sqrt{x}}$ .

chatImportant

Key point: When working with roots or reciprocals, always convert them to index form ( $x^n$ ) before applying the power rule. Functions involving roots should first be converted to fractional powers, and reciprocals should be expressed using negative powers.

Derivative notations

There are several standard ways to express derivatives in mathematics. Each notation emphasizes a different aspect of differentiation and is suited to different contexts. Understanding all notations helps you recognise derivatives in different problems and communicate your solutions clearly.

Let's explore what each notation means:

$f'(x)$ notation: This represents the derivative as a new function. When you differentiate $f(x)$ , you create a new function called f prime of x, written as $f'(x)$ . This notation is useful when thinking about the derivative as a function in its own right.

$\frac{dy}{dx}$ notation: This emphasises the derivative as a rate of change. It shows how $y$ changes with respect to changes in $x$ . The notation comes from the concept of infinitesimally small changes: $dy$ represents a tiny change in $y$ , while $dx$ represents a tiny change in $x$ . This notation is particularly helpful in applications involving rates.

$y'$ notation: This is a shorthand version of $f'(x)$ , used when the function is expressed as $y$ instead of $f(x)$ . It's quick to write and commonly used in calculations.

$\frac{d}{dx}[f(x)]$ notation: This treats differentiation as an operation performed on a function. The symbol $\frac{d}{dx}$ is called the derivative operator, and it acts on whatever function follows in the brackets. This notation is useful when you want to emphasise that differentiation is an action being applied.

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The choice of notation often depends on the context. For instance, if $y = x^2$ , you can express its derivative as:

$f'(x) = 2x$ (if $f(x) = x^2$ )
$\frac{dy}{dx} = 2x$
$y' = 2x$
$\frac{d}{dx}(x^2) = 2x$

All four expressions mean exactly the same thing.

Worked example: Expressing derivatives in different notations

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Worked Example: Using Different Notation Styles

Given the function $f(x) = x^4$ , its derivative is $4x^3$ . Let's express this derivative using different notations.

Part a: Express using $f'(x)$ notation

The derivative is $f'(x) = 4x^3$ .

This shows the derivative as a new function of $x$ .

Part b: Express using $\frac{dy}{dx}$ notation, assuming $y = f(x)$

If $y = x^4$ , then the derivative is $\frac{dy}{dx} = 4x^3$ .

This notation emphasises the rate of change of $y$ with respect to $x$ .

Part c: Express using $\frac{d}{dx}$ operator notation

The statement is $\frac{d}{dx}(x^4) = 4x^3$ .

This shows the derivative operator $\frac{d}{dx}$ being applied to the function $x^4$ .

Rules for differentiation

To differentiate more complex functions efficiently, we use additional rules that work alongside the power rule. These rules allow us to differentiate functions term by term, making polynomial functions straightforward to handle.

Constant multiple rule

When a function is multiplied by a constant, you can factor out that constant and differentiate the function separately.

The constant multiple rule states:

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

where:

$k$ is a constant
$f(x)$ is a differentiable function

In words: the derivative of a constant multiplied by a function equals the constant multiplied by the derivative of the function.

For example, if you need to differentiate $5x^3$ , you can factor out the $5$ and differentiate $x^3$ to get $5 \times 3x^2 = 15x^2$ .

Sum and difference rule

When differentiating a sum or difference of functions, you can differentiate each term separately and then add or subtract the results.

The sum and difference rule states:

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

where:

$f(x)$ and $g(x)$ are differentiable functions

In words: the derivative of a sum or difference equals the sum or difference of the individual derivatives.

For example, to differentiate $x^3 + x^2$ , differentiate each term separately: the derivative of $x^3$ is $3x^2$ , and the derivative of $x^2$ is $2x$ , so the complete derivative is $3x^2 + 2x$ .

chatImportant

Together, the power rule, constant multiple rule, and sum and difference rule allow you to differentiate any polynomial function by working through it term by term.

Worked example: Differentiating a polynomial function

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Worked Example: Polynomial Differentiation

Part a: Differentiate $f(x) = 4x^3 - 5x^2 + 7x - 10$

We'll differentiate each term separately using the power rule, and apply the constant multiple rule where needed.

Write the function: $f(x) = 4x^3 - 5x^2 + 7x - 10$

Differentiate each term: $f'(x) = \frac{d}{dx}(4x^3) - \frac{d}{dx}(5x^2) + \frac{d}{dx}(7x) - \frac{d}{dx}(10)$

Apply the power rule, constant multiple rule, and note that the derivative of a constant is zero: $f'(x) = 4 \times (3x^2) - 5 \times (2x^1) + 7 - 0$

Evaluate each term: $f'(x) = 12x^2 - 10x + 7$

Part b: Differentiate $y = 2x^4 + 3\sqrt{x} - \frac{5}{x^2}$

First, rewrite any roots or fractions in index form, then differentiate term by term.

Write the function: $y = 2x^4 + 3\sqrt{x} - \frac{5}{x^2}$

Rewrite using index form (recall $\sqrt{x} = x^{\frac{1}{2}}$ and $\frac{1}{x^2} = x^{-2}$ ): $y = 2x^4 + 3x^{\frac{1}{2}} - 5x^{-2}$

Differentiate term by term: $\frac{dy}{dx} = \frac{d}{dx}(2x^4) + \frac{d}{dx}\left(3x^{\frac{1}{2}}\right) - \frac{d}{dx}(5x^{-2})$

Apply the power rule to each term: $\frac{dy}{dx} = 2 \times (4x^3) + 3 \times \left(\frac{1}{2}x^{-\frac{1}{2}}\right) - 5 \times (-2x^{-3})$

Evaluate each term: $\frac{dy}{dx} = 8x^3 + \frac{3}{2}x^{-\frac{1}{2}} + 10x^{-3}$

The derivative is $8x^3 + \frac{3}{2}x^{-\frac{1}{2}} + 10x^{-3}$ .

Alternatively, this can be written with positive indices and surds as: $\frac{dy}{dx} = 8x^3 + \frac{3}{2\sqrt{x}} + \frac{10}{x^3}$

Simplifying before differentiation

Sometimes it's easier to simplify or expand a function before differentiating. This approach can make the differentiation process more straightforward and help avoid errors.

Worked example: Expanding brackets first

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Worked Example: Simplifying Before Differentiation

Part a: Differentiate $f(x) = (x - 3)(x^2 + 4)$

Rather than using a product rule (which you'll learn later), expand the brackets first to express the function as a polynomial.

Write the function: $f(x) = (x - 3)(x^2 + 4)$

Expand the brackets: $f(x) = x(x^2 + 4) - 3(x^2 + 4)$

$f(x) = x^3 + 4x - 3x^2 - 12$

Rewrite in descending powers of $x$ : $f(x) = x^3 - 3x^2 + 4x - 12$

Now differentiate term by term: $f'(x) = \frac{d}{dx}(x^3) - \frac{d}{dx}(3x^2) + \frac{d}{dx}(4x) - \frac{d}{dx}(12)$

Apply the differentiation rules: $f'(x) = 3x^2 - 3(2x) + 4 - 0$

Simplify: $f'(x) = 3x^2 - 6x + 4$

Part b: Differentiate $f(x) = \frac{x^2 - x}{x}$

Split the fraction into separate terms, then simplify and differentiate.

Write the function: $f(x) = \frac{x^2 - x}{x}$

Split the fraction: $f(x) = \frac{x^2}{x} - \frac{x}{x}$

Simplify each term: $f(x) = x - 1$

Now differentiate: $f'(x) = \frac{d}{dx}(x) - \frac{d}{dx}(1)$

Apply the differentiation rules: $f'(x) = 1 - 0$

Simplify: $f'(x) = 1$

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Exam tip: Always look for opportunities to simplify expressions before differentiating. Expanding brackets, splitting fractions, and converting to index form often make differentiation much easier and reduce the chance of making mistakes.

Remember!

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Key Points to Remember:

The power rule is your fundamental tool for differentiation: if $f(x) = x^n$ , then $f'(x) = nx^{n-1}$ for any real number $n$ .
Always convert roots to fractional powers and reciprocals to negative powers before applying the power rule.
You can express derivatives using multiple notations: $f'(x)$ , $\frac{dy}{dx}$ , $y'$ , or $\frac{d}{dx}[f(x)]$ - they all mean the same thing but emphasise different aspects.
The constant multiple rule and sum/difference rule let you differentiate polynomials term by term: factor out constants and differentiate each term separately.
Simplifying expressions before differentiating often makes the process easier - expand brackets, split fractions, and combine like terms when appropriate.

Derivative Notation and Basic Rules (HSC SSCE Mathematics Advanced): Revision Notes