Differentiation

Overview

Differentiation turns a curve into information. It tells you the gradient at an exact point, whether a quantity is increasing or decreasing, and where a maximum or minimum occurs. In WJEC A-Level Mathematics 2.1.7, candidates are expected to move confidently between a graph, a derivative and a practical model. A one-line error in the derivative can therefore cost several later method marks.

This topic is central to calculus and returns later in integration, kinematics, modelling and numerical methods. Typical WJEC questions ask candidates to differentiate a polynomial, use first principles for a small positive integer power, find a tangent or normal, classify stationary points, sketch from derivative information, or maximise/minimise a simple quantity. The key habit is simple: differentiate first, then interpret what the derivative means in the question.

Specification focus: WJEC 2.1.7 requires the derivative as a tangent gradient and rate of change, gradient-function sketches, second derivatives, first principles for small positive integer powers, rational powers, tangents/normals, stationary points, increasing/decreasing intervals, simple curve sketches and simple optimisation.

Audio Revision Podcast

10-minute tutor-led recap: listen after reading the overview, then pause during the quick-fire recall section to test retrieval.

Key Concepts

1. The derivative: an instant slope and a rate of change

For a curve y=f(x), the derivative f'(x) or \frac{dy}{dx} gives the gradient of the tangent at a general point. The word general matters: f'(x) is an expression in x, not yet a single number. Once a particular value, say x=2, is substituted, f'(2) is the gradient at that point.

Imagine a hilly road. Average gradient compares two places on the road. The derivative is the steepness exactly where the bicycle is now. In a modelling question, this becomes a rate: if s is displacement and t is time, \frac{ds}{dt} is velocity. The same mathematics works because both gradient and rate describe change per unit change.

Exam wording to decode

Worked micro-example: If f(x)=x^3-2x, then f'(x)=3x^2-2. At x=2, the tangent gradient is 3(2)^2-2=10. Candidates who write just “10” when asked for the derivative lose credit: the derivative is 3x^2-2; 10 is one value of it.

2. Differentiation from first principles

First principles explains why differentiation gives a gradient. Start with the gradient of a chord through (x,f(x)) and (x+h,f(x+h)):

\frac{f(x+h)-f(x)}{h}

The chord is not yet a tangent. Let h approach zero. The second point moves into the first point, so the chord gradient tends to the tangent gradient:

f'(x)=\lim_{h\to0}\frac{f(x+h)-f(x)}{h}.

For WJEC 2.1.7, practise this securely for small positive integer powers, up to and including x^3. The mark-winning sequence is always: substitute x+h, expand, subtract f(x), factor out h, cancel h, then take the limit. You cannot substitute h=0 before cancellation because that would create division by zero.

Why it works: expansion exposes a common factor of h, representing the shrinking horizontal gap. After cancellation, the remaining expression has a sensible limit. This is the rigorous route from an average rate of change to an instantaneous rate of change.

Memory hook: S-E-S-F-C-L: Substitute, Expand, Subtract, Factor, Cancel, Limit. Write those initials in the margin before a first-principles question.

3. The power rule, including rational powers

For a term ax^n:

\frac{d}{dx}(ax^n)=anx^{n-1}.

Multiply by the existing power, then reduce the power by one. Constants differentiate to zero. Differentiate each term separately in a sum or difference.

Do not confuse differentiation with integration. Differentiation makes the power one smaller; integration makes it one larger and divides by the new power. Before differentiating, rewrite roots and fractions as powers where useful, for example \sqrt[3]{x^2}=x^{2/3} and \frac{4}{x^2}=4x^{-2}.

Examiner commentary: Credit is normally awarded for a correct derivative before a later substitution. Therefore state f'(x)= clearly on its own line. This protects method marks and makes an arithmetic slip easier to spot.

4. Tangents and normals

A tangent touches the curve at the stated point and has gradient m=f'(a). A normal is perpendicular to that tangent. Perpendicular non-vertical lines have gradients whose product is -1, so:

m_{\text{normal}}=-\frac{1}{m_{\text{tangent}}}.

Use point-slope form once both a point and a gradient are known:

y-y_1=m(x-x_1).

Reliable method for a tangent or normal

1. Find the coordinate on the curve: (a,f(a)).
2. Differentiate and calculate f'(a) for the tangent gradient.
3. For a tangent, use that gradient. For a normal, take its negative reciprocal.
4. Substitute the point and chosen gradient into point-slope form.
5. Simplify only after the correct line has been formed.

A frequent error is to negate a gradient but not take the reciprocal. If the tangent gradient is -5, the normal gradient is \frac15, not 5 or -\frac15.

5. Stationary points, second derivatives and monotonicity

A stationary point is where the tangent is horizontal, so f'(x)=0. It may be a local maximum, a local minimum or a stationary point of inflection. Finding f'(x)=0 gives only the possible x-coordinates. Substitute each value into f(x) to find the corresponding y-coordinate.

The second derivative measures how the gradient is changing:

• f''(a)>0: gradient is increasing, the curve bends like a smile, so a local minimum.
• f''(a)<0: gradient is decreasing, the curve bends like a frown, so a local maximum.
• f''(a)=0: this test is inconclusive. Test the sign of f'(x) on either side or use a justified sketch.

For increasing/decreasing intervals, do not merely list stationary points. Solve the inequality. If f'(x)>0, f is increasing; if f'(x)<0, f is decreasing. A sign diagram is often the quickest way to make this visible.

6. Optimisation: mathematics with a decision

Optimisation questions are a chain of reasoning, not a request to differentiate a random expression. First make one variable dependent on the other using the constraint. Then write the quantity to be maximised or minimised in one variable. Differentiate, solve \frac{dQ}{dx}=0, classify the result, and state it in context with units.

For example, a rectangular pen against a wall needs fencing on three sides only. If 40 m of fencing is available and each width is x, the length is 40-2x. The area is A=x(40-2x), not just 40x. The constraint has changed the model. The second derivative, a sign change, or an end-point check supplies the justification for “maximum” or “minimum”.

Examiner habit: in a 5- or 6-mark optimisation question, candidates should expect credit for the model, derivative, stationary condition, classification and contextual conclusion. A naked value of x rarely earns full marks.

Mathematical Relationships and Formula Status

Graph and Data Skills

Sketches should communicate derivative information, not artwork. Mark stationary points with their coordinates where known. Use the sign of f'(x) to decide whether each section rises or falls and use f''(x) to comment on curvature. A graph of f'(x) is itself a gradient map: where it is above the x-axis, f increases; where it crosses the x-axis, f has a stationary point; where it is below, f decreases. Its own gradient is f''(x).

Practical Applications

Differentiation drives decisions whenever a changing quantity matters: a designer can maximise area from fixed materials; a transport model uses velocity as the rate of change of displacement; a business model can identify a maximum revenue point. The exam may use an unfamiliar context, but the marks still reward the same sequence: form a function, differentiate, interpret, justify.## Examiner’s Final Check

Before moving on, run this 20-second audit. Have you differentiated rather than integrated? Have you substituted into the original function for a coordinate, not into the derivative? If a line is requested, does it pass through the point and use the correct tangent or normal gradient? If an optimum is claimed, have you justified it and answered in the real-world units? These checks target the small omissions that turn an otherwise correct solution into dropped marks.