Differentiation of polynomials and tangent lines (QCE Math Methods Unit 2)

What this dot point is asking

QCAA wants you to differentiate polynomial functions using the three core rules (power, sum, constant multiple), and to use the derivative as the gradient function for tangent and normal line problems.

The three rules

Power rule. For any rational $n$:

This was proved from first principles for positive integer $n$ in the previous dot point. The same form holds for $n$ rational or negative, with the standard domain restrictions.

Sum (and difference) rule. The derivative of a sum is the sum of the derivatives:

Constant multiple rule. A constant pulls outside the derivative:

Constant function. The derivative of a constant is zero:

Combined, these let you differentiate any polynomial term by term.

Why the rules let you differentiate any polynomial

A polynomial is a sum of constant multiples of powers of $x$, so the constant-multiple rule handles each coefficient, the power rule handles each power, and the sum rule combines them. That is why differentiation of a polynomial is purely mechanical: apply the power rule to every term, multiply by its coefficient, and add. The same three rules underpin the calculus of more complicated functions in Year 12, where they combine with the product, quotient and chain rules.

Standard manipulations

Coefficient times power

$\dfrac{d}{dx} (5 x^3) = 5 \cdot 3 x^2 = 15 x^2$.

Negative power

$\dfrac{d}{dx} (x^{-2}) = -2 x^{-3} = -2/x^3$.

Rational power

$\dfrac{d}{dx} (x^{1/2}) = \tfrac{1}{2} x^{-1/2} = 1/(2 \sqrt x)$.

Combine terms before differentiating

Expand brackets and split fractions if needed. For $\dfrac{x^3 - 2x}{x}$, first simplify to $x^2 - 2$, then differentiate to $2x$.

Higher derivatives

Differentiating a polynomial gives another polynomial, which can itself be differentiated. The second derivative $f''(x)$ measures the rate of change of the gradient and describes concavity: $f''(x) > 0$ means the curve is concave up (holding water) and $f''(x) < 0$ means concave down. For $f(x) = 2x^3 - 5x^2 + 4x - 1$, the first derivative is $6x^2 - 10x + 4$ and the second is $12x - 10$. Second derivatives become important when classifying stationary points and analysing motion, where they represent acceleration.

Tangent and normal lines

The tangent to $y = f(x)$ at $x = a$:

• Point: $(a, f(a))$.
• Gradient: $m = f'(a)$.
• Equation: $y - f(a) = f'(a) (x - a)$.

The normal at the same point:

• Perpendicular to the tangent.
• Gradient: $-1/f'(a)$ (provided $f'(a) \neq 0$).
• Equation: $y - f(a) = -\dfrac{1}{f'(a)} (x - a)$.

If $f'(a) = 0$ the tangent is horizontal and the normal is vertical ($x = a$).

Rewriting before differentiating

The power rule applies cleanly only to terms in the form $cx^n$, so an expression must first be put into that shape. Expand any brackets, split fractions over a common denominator, and rewrite roots and reciprocals as powers: $\sqrt x$ becomes $x^{1/2}$, and $\dfrac{1}{x^2}$ becomes $x^{-2}$. Skipping this step is the most common reason a derivative comes out wrong, because the power rule cannot be applied to a product or quotient directly at this stage of the course.