Introduction to differential calculus, including the gradient at a point, the derivative as a function, and the power rule for derivatives of polynomial functions

What this dot point is asking

QCAA wants Year 11 students to be introduced to differential calculus through the gradient concept, to define the derivative as a function, to apply the power rule to polynomials, and to classify stationary points using the first derivative. This is the gateway to all the calculus of Units 3 and 4.

The gradient and derivative

The average gradient of $f$ between two points is the slope of the secant joining them, $\dfrac{f(b) - f(a)}{b - a}$. The instantaneous gradient at $x = a$ is the limit of average gradients as the second point slides toward the first, $\displaystyle\lim_{h \to 0}\dfrac{f(a + h) - f(a)}{h}$, which equals $f'(a)$ and is the slope of the tangent line at that point.

Crucially, the derivative $f'(x)$ is itself a function: it takes any input $x$ and returns the gradient of the tangent there. This shift from "the gradient at one point" to "a gradient function for every point" is the central idea of differential calculus, and it is what makes the power rule so useful, since it gives the whole gradient function in one step.

Power rule

The power rule states that for $f(x) = x^n$, the derivative is $f'(x) = nx^{n-1}$: bring the index down as a multiplier and reduce it by one. It holds for any real index, including negative and fractional ones once those are introduced. The rule extends to polynomials through linearity, $\dfrac{d}{dx}[af(x) + bg(x)] = af'(x) + bg'(x)$, so each term is differentiated separately. A constant has zero gradient, so its derivative is $0$, and a constant multiple is preserved, $\dfrac{d}{dx}[cf(x)] = c f'(x)$.

Increasing and decreasing intervals

The sign of the derivative describes where a function rises or falls: $f'(x) > 0$ on an interval means $f$ is increasing there, and $f'(x) < 0$ means it is decreasing. The boundaries between these intervals are the stationary points, where $f'(x) = 0$. Building a sign diagram of $f'$ is therefore the systematic way to find and classify turning points and to sketch the overall shape of a polynomial, a method used throughout the calculus content that follows.

Tangents and normals

A tangent line touches the curve at one point and shares its gradient there, so the tangent at $x = a$ passes through $(a, f(a))$ with gradient $f'(a)$, giving $y - f(a) = f'(a)(x - a)$. The normal is perpendicular to the tangent at the same point, so its gradient is the negative reciprocal $-\dfrac{1}{f'(a)}$ (and the normal is vertical, $x = a$, when the tangent is horizontal). These two lines are a recurring application of the derivative and connect the algebra of gradients to the geometry of the curve.

Stationary points

A stationary point occurs where the tangent is horizontal, that is where $f'(x) = 0$. To classify it, examine the sign of $f'$ immediately on each side:

• Positive then negative: the function rises then falls, a local maximum.
• Negative then positive: the function falls then rises, a local minimum.
• The same sign on both sides: a stationary point of inflection (the curve flattens momentarily without turning).

This first-derivative test is the standard QCE method, and presenting a clear sign diagram of $f'$ is what earns the classification marks. Stationary points are the basis of optimisation and curve sketching met later in the course.