Apply the power rule, the sum rule, and the constant-multiple rule to differentiate polynomial functions, and use the derivative to find tangent and normal line equations

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.

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 $2x$.

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 ($x = a$).

Worked example

Find the equation of the tangent to $y = x^3 - 4x$ at the point where $x = 1$.

$f(1) = 1 - 4 = -3$. Point: $(1, -3)$.

$f'(x) = 3x^2 - 4$. $f'(1) = 3 - 4 = -1$.

Tangent: $y - (-3) = -1 (x - 1)$, so $y = -x - 2$.

Common traps

Forgetting to drop the power by one. $\dfrac{d}{dx} x^4 = 4 x^3$, not $4x^4$.

Misapplying the rule to a constant times $x$. $\dfrac{d}{dx} (3x) = 3$, not $3x$ or $3 x^0$ stated awkwardly.

Confusing tangent with secant. A secant joins two points on the curve; a tangent touches at one. The derivative gives the tangent gradient.

Forgetting the negative-reciprocal rule for normals. $m_{\text{normal}} = -1/m_{\text{tangent}}$. A common slip is using $-m_{\text{tangent}}$ or $1/m_{\text{tangent}}$.

In one sentence

Differentiate any polynomial term by term using the power rule $\dfrac{d}{dx} x^n = n x^{n-1}$ together with the sum and constant-multiple rules; the derivative at $x = a$ is the gradient of the tangent line, and the normal at the same point has gradient $-1/f'(a)$.