Tangents & Normals

The tangent at a point on a curve has gradient $\frac{dy}{dx}$ evaluated there; the normal is perpendicular to it. One routine covers both, and the only real hazard is answering the wrong one.

The routine

Find the equation of the normal to $y = x^3 - 4x + 1$ at the point where $x = 2$.

1. Point: $y(2) = 8 - 8 + 1 = 1 \to (2, 1)$
2. Differentiate: $\frac{dy}{dx} = 3x^2 - 4$
3. Substitute: at $x = 2$, gradient of tangent $m = 8$
4. Tangent or normal? Normal asked $\to$ gradient $=$ $-\frac{1}{8}$ (negative reciprocal, stated)
5. Build the line: $y - 1 = -\frac{1}{8}(x - 2) \to$ $8y + x = 10$ (point-gradient form)

Five visible steps, each a marking point. The examiner-report perennial: students execute steps 1–3 perfectly, then build the tangent when the normal was asked (or vice versa). Underline the word in the question, and write step 4 explicitly even when the tangent is wanted (“tangent gradient $= 8$”), the stated decision is cheap insurance.

The variants

• “Find where the tangent is parallel to $y = 5x - 2$”: set $\frac{dy}{dx} = 5$, solve for $x$, gradient condition, parallel means equal gradients
• “The tangent at $P$ passes through the origin”: build the tangent with unknown contact point, impose the through-origin condition, harder, but the same five steps with algebra carried symbolically
• “Where does the normal meet the curve again?”: build the normal, then solve simultaneously with the curve
• With product/quotient/chain functions: step 2 invokes the differentiation rules, the routine is unchanged

Tangents to circles need no calculus, radius $\perp$ tangent does it (circle tangents); reaching for $\frac{dy}{dx}$ there is legal but slow.

Common mistakes

• Tangent built for normal (and vice versa), the topic’s defining error
• The $y$-coordinate never computed (line built through $(2, 0)$ by reflex)
• Gradient function used as the gradient (unsubstituted $3x^2 - 4$ in the line equation)
• Final form ignored when the question specified one
• The $-\frac{1}{m}$ flip applied to the normal of a normal (double-flipping back to the tangent)

Full topic context: Calculus notes · the exam drill: differentiation technique.