Derivatives of Basic Functions Revision Notes for HSC SSCE Mathematics Advanced

Derivatives of Basic Functions

Introduction

In calculus, two fundamental types of functions have straightforward derivative rules: constant functions and linear functions. Understanding these rules becomes clearer when we remember that the derivative represents the gradient (or slope) of a function's graph. These basic rules form the foundation for differentiating more complex functions.

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The concept of a derivative as a gradient is crucial—it connects the algebraic process of differentiation with the geometric interpretation of slope on a graph.

Constant functions

Understanding constants

A constant is a number that remains unchanged. For instance, in the expression $x + 11$ , the number $11$ is a constant because its value never varies.

A constant function is a function that produces only one output value, regardless of the input. This means no matter what $x$ -value you choose, the function always returns the same result.

The form and graph of constant functions

A constant function can be written as $f(x) = c$ , where $c$ represents a constant value. When we graph the equation $y = c$ , we get a horizontal line. This horizontal line is significant because it has a gradient of zero—there is no vertical change as we move along the line, so there is no slope.

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Critical Connection: Since horizontal lines have zero gradient, the derivative of any constant function must equal zero. This is not just a rule to memorize—it's a direct consequence of what derivatives measure!

The derivative rule for constant functions

Since the graph of a constant function is always a horizontal line with zero gradient, the derivative must equal zero. This gives us the important rule:

If $f(x) = c$ (where $c$ is any constant), then $f'(x) = 0$ .

This means the rate of change of a constant function is zero, which makes intuitive sense—the function isn't changing at all!

Linear functions

Understanding linear functions

A linear function describes a relationship where $y$ depends on $x$ according to the equation $y = mx + c$ . In this equation, both $m$ and $c$ are constants. The graph of a linear function is a straight line, where $m$ represents the gradient (the steepness of the line) and $c$ represents the y-intercept (the point where the line crosses the y-axis).

The derivative rule for linear functions

The key property of a straight line is that it has a constant gradient. This gradient never changes as you move along the line—it remains the same value $m$ everywhere. Because the derivative measures the gradient, and the gradient is constantly $m$ , the derivative is simply this value:

If $f(x) = mx + c$ , then $f'(x) = m$ .

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Why does the constant disappear?

Notice that the constant term $c$ disappears when we differentiate. This happens because $c$ affects the position of the line (via the y-intercept) but does not affect its gradient. The derivative only captures the rate of change, which is determined entirely by $m$ .

Special cases of linear functions

Two special cases of linear functions are particularly useful:

If $f(x) = x$ (where $m = 1$ and $c = 0$ ), then $f'(x) = 1$

This represents a line passing through the origin with gradient $1$ .
If $f(x) = mx$ (where $c = 0$ ), then $f'(x) = m$

This represents a line passing through the origin with gradient $m$ .

These special cases confirm that when there is no constant term, we still simply get the coefficient of $x$ as our derivative.

Worked examples

Let's apply these rules to find derivatives of specific functions.

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Worked Example 1a: Derivative of a Constant Function

Find the derivative of $f(x) = 15$

Strategy: Identify this as a constant function and use the rule that states the derivative of any constant equals zero.

Working:

$f(x) = 15$

$f'(x) = 0$

The derivative is zero because $15$ is a constant value.

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Worked Example 1b: Derivative of a Linear Function (No Constant Term)

Find the derivative of $g(x) = -3x$

Strategy: Identify this as a linear function of the form $y = mx$ (with no constant term) and use the rule that states the derivative equals $m$ .

Working:

$g(x) = -3x$

$g'(x) = -3$

We apply the linear function rule where $m = -3$ , so the derivative is $-3$ .

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Worked Example 1c: Derivative of a Linear Function (With Constant Term)

Find the derivative of $h(x) = 4x - 9$

Strategy: Identify this as a linear function of the form $y = mx + c$ and use the rule that states the derivative equals $m$ .

Working:

$h(x) = 4x - 9$

$h'(x) = 4$

We apply the linear function rule where $m = 4$ . The constant term $-9$ disappears during differentiation.

Therefore, the derivative of $h(x) = 4x - 9$ is $h'(x) = 4$ .

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Key Points to Remember:

The derivative of any constant function is always zero: if $f(x) = c$ , then $f'(x) = 0$ . This is because horizontal lines have zero gradient.
The derivative of a linear function equals its gradient: if $f(x) = mx + c$ , then $f'(x) = m$ . The constant term $c$ vanishes during differentiation.
When differentiating, constants disappear because they don't contribute to the rate of change—only the coefficient of $x$ matters for determining the gradient.
For the special case $f(x) = x$ , the derivative is simply $f'(x) = 1$ , representing a line with gradient $1$ .
These basic rules connect directly to the graphical interpretation: derivatives measure gradients, so a horizontal line (constant function) has derivative $0$ , and a slanted line (linear function) has derivative equal to its slope.

Derivatives of Basic Functions (HSC SSCE Mathematics Advanced): Revision Notes