Differentiation in N(A)-Level Additional Mathematics (4051): the derivative as a gradient, the power, chain, product and quotient rules, tangents and normals, and stationary points and their nature

From gradient to powerful tool

Differentiation is the heart of the Calculus strand of N(A)-Level Additional Mathematics (SEAB 4051). It starts from a single idea, the gradient of a curve at a point, and grows into a toolkit for finding tangents, normals and turning points. The student who treats differentiation as the gradient machine rather than a set of disconnected formulae handles the applications far more confidently.

This module covers three outcomes that build in sequence: the rules of differentiation, tangents and normals, and stationary points and their nature. This overview ties the three dot points together; each has its own worked answers and practice.

The rules of differentiation

The differentiation rules outcome establishes what the derivative means and how to compute it. The power rule is $\dfrac{d}{dx}(x^n) = n x^{n-1}$, and you extend it with the chain rule $\dfrac{dy}{dx} = \dfrac{dy}{du}\cdot\dfrac{du}{dx}$, the product rule $(uv)' = u'v + uv'$, and the quotient rule $\left(\dfrac{u}{v}\right)' = \dfrac{u'v - uv'}{v^2}$. The skill is recognising the structure of an expression and choosing the right rule.

Tangents and normals

The tangents and normals outcome applies the derivative as a gradient. The tangent at a point has gradient $\dfrac{dy}{dx}$ evaluated there, and the normal is perpendicular, so its gradient is the negative reciprocal. Both lines come from $y - y_1 = m(x - x_1)$ with the appropriate gradient.

Stationary points and their nature

The stationary points and their nature outcome locates points where $\dfrac{dy}{dx} = 0$ and classifies each as a maximum or a minimum. The second derivative test reads the sign of $\dfrac{d^2 y}{dx^2}$ at the point (negative for a maximum, positive for a minimum); when it is zero you use a sign test on $\dfrac{dy}{dx}$ instead. This underlies optimisation problems.

How this module is examined

• Both papers, all questions. Paper 1 and Paper 2 (each 1 hour 45 minutes, 70 marks, 50 percent) cover the full syllabus, and you answer every question.
• State the rule you are using. For products, quotients and composites, showing $u$, $v$ and their derivatives makes the method clear and protects part marks.
• Justify the nature of a stationary point. Do not just write "maximum"; show the second derivative is negative there, or the sign change in $\dfrac{dy}{dx}$, to earn the reasoning marks.

Check your knowledge

Short questions across the three outcomes. Work them with full method, then check the solutions.

1. Differentiate $y = 3x^4 - 2x + 5$ with respect to $x$. (2 marks)
2. Differentiate $y = (2x + 1)^3$ using the chain rule. (2 marks)
3. Find the equation of the tangent to $y = x^3$ at the point $(1, 1)$. (3 marks)
4. Find and classify the stationary point of $y = x^2 - 4x + 1$. (3 marks)