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 introduce differential calculus via the gradient concept, define the derivative, apply the power rule to polynomials, and classify stationary points using the first derivative.

The gradient and derivative

Average gradient between two points: $(f(b) - f(a)) / (b - a)$.

Instantaneous gradient at $x = a$: limit as $h \to 0$ of $(f(a + h) - f(a))/h$, equals $f'(a)$.

The derivative $f'(x)$ is a function: at each $x$, it gives the gradient of the tangent.

Power rule

For $f(x) = x^n$: $f'(x) = n x^{n-1}$.

Linearity: $d/dx[a f(x) + b g(x)] = a f'(x) + b g'(x)$.

Constants: derivative of a constant is 0.

Tangents and normals

Tangent. Line at $x = a$ with gradient $f'(a)$ passing through $(a, f(a))$: $y - f(a) = f'(a)(x - a)$.

Normal. Perpendicular: gradient $-1/f'(a)$.

Stationary points

Where $f'(x) = 0$. Classify by sign of $f'$ around the point:

• Positive to negative: local maximum.
• Negative to positive: local minimum.
• Same sign on both sides: stationary inflection.

Common errors

Power rule on $e^x$, $\sin x$, $\ln x$. Power rule applies to $x^n$, not these (introduced in Unit 3).

Forgetting to set $f' = 0$ for stationary points. Stationary means zero gradient.

Tangent gradient confused with function value. $f(a)$ is value; $f'(a)$ is gradient.

In one sentence

Differential calculus is introduced in Unit 2 through the gradient at a point (limit of average gradients, equal to $f'(a)$), the derivative as a function obtained via the power rule $d/dx(x^n) = nx^{n-1}$ extended by linearity to polynomial sums, with applications to tangent lines and stationary points (found by solving $f'(x) = 0$ and classified by the sign of $f'$).