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Grade 12 Differential Calculus: How to Identify the Question Type

Differential calculus is Grade 12-only content. Questions test finding derivatives (from first principles or rules), using derivatives to find gradients and equations of tangents, and sketching cubic graphs by finding stationary points and points of inflection.

Type 1: Finding the derivative from first principles

Trigger words: "From first principles", "using the definition of the derivative"

Trigger structure: The instruction explicitly says 'first principles', do NOT use the power rule here even if it would give the same answer.

Do not confuse with: The power rule: first principles requires the limit definition. Using the rule when first principles is specified earns zero.

Method (no numbers, just the steps)

  1. Write the definition: f'(x) = lim[h→0] [f(x + h) − f(x)] / h
  2. Substitute f(x + h) by replacing every x with (x + h)
  3. Expand and simplify the numerator, the h in the denominator must cancel
  4. Take the limit as h → 0
Grade 12Paper 1Level 2, Routine Procedures
7.1 Determine f'(x) from first principles if it is given that f(x) = 4 − 7x.

Source: DBE Mathematics Paper 1, November 2019, Question 7.1

Common mistake

Not expanding f(x + h) fully before simplifying, the h must cancel exactly or you cannot take the limit.

See the progression: same type, increasing difficulty

Easy
Determine f'(x) from first principles if f(x) = 4x.

Practice question: not sourced from a past paper.

Medium
Determine f'(x) from first principles if f(x) = x² + 2.

Practice question: not sourced from a past paper.

Hard
Determine f'(x) from first principles if f(x) = 1/x.

Practice question: not sourced from a past paper.

Type 2: Differentiation rules (power rule)

Trigger words: "Determine f'(x)", "Find dy/dx", "Differentiate"

Trigger structure: A polynomial, rational expression, or surd that can be rewritten as a sum of power terms.

Do not confuse with: First principles: use the rules unless 'first principles' is specified.

Method (no numbers, just the steps)

  1. Rewrite the function so every term is of the form ax^n
  2. Rewrite fractions: 1/x = x^(−1); surds: √x = x^(1/2)
  3. Apply: d/dx[ax^n] = anx^(n−1)
  4. Simplify: rewrite negative exponents as fractions if needed
Grade 12Paper 1Level 2, Routine Procedures
7.2 Determine dy/dx if y = 4x⁸ + √(x³)

Source: DBE Mathematics Paper 1, November 2019, Question 7.2

Common mistake

Differentiating a product by differentiating each factor separately, this only works if one factor is a constant. For variable products, expand first.

See the progression: same type, increasing difficulty

Easy
Find dy/dx if y = 5x³ − 2x + 7.

Practice question: not sourced from a past paper.

Medium
Determine f'(x) if f(x) = (3x² − 1)/x.

Practice question: not sourced from a past paper.

Hard
Determine dy/dx if y = (2√x + 3)/√x.

Practice question: not sourced from a past paper.

Type 3: Gradient of a tangent and equation of a tangent

Trigger words: "Determine the gradient of the tangent", "Find the equation of the tangent at", "At which point is the gradient equal to"

Trigger structure: A curve and a specific point (or gradient value) are given. The question asks for the slope or equation of the line touching the curve at that point.

Method (no numbers, just the steps)

  1. Differentiate f(x) to get f'(x)
  2. Substitute the given x-value into f'(x) to find the gradient m
  3. Use y − y₁ = m(x − x₁) to write the tangent equation, where (x₁, y₁) is the point on the curve
Grade 12Paper 1Level 3, Complex Procedures
7.4 The curve with equation y = x + 12/x passes through the point A(2 ; b). Determine the equation of the line perpendicular to the tangent to the curve at A.

Source: DBE Mathematics Paper 1, November 2019, Question 7.4

See the progression: same type, increasing difficulty

Easy
f(x) = x² + 1. Find the gradient of f at x = 3.

Practice question: not sourced from a past paper.

Medium
The tangent to f(x) = x³ − 2x at x = 1. Determine the equation of the tangent.

Practice question: not sourced from a past paper.

Hard
f(x) = −x² + 4x. Find the x-coordinate(s) of the point(s) where the gradient of f equals 2.

Practice question: not sourced from a past paper.

Type 4: Cubic graph: stationary points and sketching

Trigger words: "Sketch the graph of", "Find the coordinates of the turning points", "Determine the x-values of the stationary points"

Trigger structure: A cubic function f(x) = ax³ + bx² + cx + d is given. You must find turning points, x-intercepts, and sketch.

Method (no numbers, just the steps)

  1. Find x-intercepts: factorise f(x), try synthetic division with rational root candidates
  2. Find stationary points: solve f'(x) = 0
  3. Determine nature: f''(x) > 0 at x₀ → local minimum; f''(x) < 0 at x₀ → local maximum
  4. Find y-values of stationary points by substituting back into f(x)
  5. Note the y-intercept: f(0)
Grade 12Paper 1Level 3, Complex Procedures
g(x) = x³ + x² − 16x + 20 9.1.2 Determine the coordinates of the turning points of g.

Source: DBE Mathematics Paper 1, November 2018, Question 9.1.2

Common mistake

Finding the x-values of the turning points from f'(x) = 0 but forgetting to substitute back into f(x), the question asks for coordinates, not just x-values.

See the progression: same type, increasing difficulty

Easy
f(x) = x³ − 6x. Find the x-values of the stationary points.

Practice question: not sourced from a past paper.

Medium
f(x) = −x³ + 3x + 2. Determine the coordinates of the local maximum.

Practice question: not sourced from a past paper.

Hard
f(x) = x³ − x² − 8x + 12. Determine the x-intercepts and stationary points, then sketch f.

Practice question: not sourced from a past paper.

Type 5: Applied calculus (optimisation and rate of change)

Trigger words: "Maximum volume", "minimum cost", "rate of change", "At what time is the velocity zero"

Trigger structure: A real-world scenario gives a formula. The question asks for a maximum or minimum value, or asks how fast something is changing.

Method (no numbers, just the steps)

  1. Write the quantity to optimise as a function of one variable
  2. Differentiate and set f'(x) = 0 to find the critical value
  3. Verify it is a maximum or minimum using f''(x) or by checking function values
  4. For rate of change: differentiate and substitute the given value of x or t
Grade 12Paper 1Level 4, Problem Solving
After flying a short distance, an insect came to rest on a wall. Thereafter the insect started crawling on the wall. The path that the insect crawled can be described by h(t) = (t − 6)(−2t² + 3t − 6), where h is the height (in cm) above the floor and t is the time (in minutes) since the insect started crawling. 8.3 Determine the maximum height that the insect reached above the floor.

Source: DBE Mathematics Paper 1, November 2019, Question 8.3

See the progression: same type, increasing difficulty

Easy
The height of a ball thrown upwards is h(t) = 20t − 5t². At what time does the ball reach maximum height?

Practice question: not sourced from a past paper.

Medium
A rectangle has a perimeter of 40 m. Find the dimensions that maximise the area.

Practice question: not sourced from a past paper.

Hard
A box without a lid is made from a 12 cm × 12 cm sheet of cardboard by cutting equal squares of side x from each corner and folding up the sides. Determine the value of x that maximises the volume of the box.

Practice question: not sourced from a past paper.

Words like determine and hence appear throughout this topic, see the instruction word glossary for full definitions.