The derivative from first principles (QCE Math Methods Unit 2)

What this dot point is asking

QCAA wants you to define the derivative as the limit of an average rate of change and apply the first-principles definition to differentiate polynomial functions. This is the foundation for every shortcut rule (power rule, sum rule, constant multiple) that follows.

Average and instantaneous rates of change

The average rate of change of $f$ between $x = a$ and $x = a + h$ is the slope of the secant line:

The instantaneous rate of change at $x = a$ is the limit of this average as $h \to 0$. This is the slope of the tangent line at $a$ and is called the derivative.

The derivative as a limit

The derivative of $f$ at $x$ is:

provided the limit exists. Other notations: $\dfrac{dy}{dx}$, $\dfrac{d}{dx} f(x)$, $Df(x)$.

First-principles procedure

Four standard steps:

1. Write $f(x + h)$ by substituting $x + h$ into the function.
2. Compute $f(x + h) - f(x)$ and simplify.
3. Divide by $h$ and simplify so that no $h$ appears in the denominator.
4. Take the limit as $h \to 0$ by direct substitution.

The key algebraic move is to factor $h$ out of every term in the numerator of step 2; then it cancels with the denominator in step 3, leaving a polynomial in $x$ and $h$. Step 4 then sends $h$ to zero.

The basic results

For $f(x) = c$: $f'(x) = 0$ (constant function has zero slope everywhere).

For $f(x) = x$: $f'(x) = 1$.

For $f(x) = x^2$: $f'(x) = 2x$.

For $f(x) = x^n$ with $n$ a positive integer: $f'(x) = n x^{n-1}$ (the power rule, proved by binomial expansion in first principles).

Why the limit is necessary

The difference quotient $\dfrac{f(x + h) - f(x)}{h}$ is the gradient of the secant line joining two points on the curve. As $h$ shrinks, the second point slides toward the first and the secant rotates toward the tangent. You cannot simply set $h = 0$, because that gives $\dfrac{0}{0}$, which is undefined; the limit is what makes the idea rigorous. Algebraically, simplifying first removes the $h$ in the denominator so that substituting $h = 0$ is then legitimate, which is the whole reason the four-step procedure works.

The geometric meaning

The value $f'(a)$ is the gradient of the tangent line to $y = f(x)$ at $x = a$, and equivalently the instantaneous rate of change of $f$ there. A positive derivative means the function is increasing, a negative derivative means decreasing, and a zero derivative marks a stationary point. This geometric reading is what connects first principles to the later study of stationary points, optimisation and curve sketching, where the derivative is the central tool.

How this appears in assessment

In IA1 a four-mark first-principles question on a polynomial up to degree three is standard, and the marker's focus is the procedure (difference quotient, factor, cancel, limit) as much as the final answer. In the external assessment, Paper 1 may ask you to identify the difference quotient or differentiate a simple polynomial from first principles, while Paper 2 builds on the result to introduce the power rule and combined-rule applications in Unit 3. Showing every step, and not collapsing the limit prematurely, is what secures full marks.