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  1. Grade 12 Mathematics/

Differential Calculus

Table of Contents

Calculus: The Logic of Change (~35 marks, Paper 1)
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Differential Calculus is worth 35 marks in Paper 1 — tied with Functions as the heaviest topic. It builds directly on your algebra and functions knowledge.

The Big Idea: Instantaneous Rate of Change
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Imagine you are in a car:

  • Average Speed: You drove 100 km in 1 hour. Your speed was 100 km/h on average.
  • Instant Speed: You look at the speedometer right now. It says 110 km/h.

Calculus is the math of the speedometer. It calculates the gradient (speed) at an exact point, rather than over a range. The tool that does this is called the derivative.

The Key Concepts at a Glance
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ConceptWhat it meansFormula/Method
First principlesThe formal definition of the derivative using limits$f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}$
Power ruleThe shortcut for differentiatingIf $f(x) = ax^n$, then $f'(x) = nax^{n-1}$
Tangent lineA line that touches the curve at exactly one pointGradient = $f'(a)$; equation via point-gradient form
Stationary pointsWhere the gradient is zero ($f'(x) = 0$)Turning points of the graph
Point of inflectionWhere concavity changes ($f''(x) = 0$)The “middle” of a cubic’s S-shape
OptimizationFinding max/min values in real-world contextsSet $f'(x) = 0$, solve, check with $f''(x)$

The Differentiation Rules You Must Know
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FunctionDerivativeExample
$f(x) = ax^n$$f'(x) = nax^{n-1}$$f(x) = 3x^4 \Rightarrow f'(x) = 12x^3$
$f(x) = c$ (constant)$f'(x) = 0$$f(x) = 7 \Rightarrow f'(x) = 0$
$f(x) = ax$$f'(x) = a$$f(x) = 5x \Rightarrow f'(x) = 5$
Sum/DifferenceDifferentiate term by term$f(x) = x^3 - 2x \Rightarrow f'(x) = 3x^2 - 2$

⚠️ Before differentiating: You MUST rewrite the expression so every term is in the form $ax^n$. This means: expand brackets, split fractions, convert roots to powers. You CANNOT differentiate a product or quotient directly in Grade 12.

Cubic Graphs: The Sketching Checklist
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For $f(x) = ax^3 + bx^2 + cx + d$:

  1. $y$-intercept: Let $x = 0$ → $f(0) = d$
  2. $x$-intercepts: Solve $f(x) = 0$ (use Factor Theorem)
  3. Stationary points: Solve $f'(x) = 0$ → gives $x$-values of turning points
  4. Nature of stationary points: Use $f''(x)$ — positive = local min, negative = local max
  5. Point of inflection: Solve $f''(x) = 0$
  6. End behaviour: $a > 0$: rises right, falls left. $a < 0$: falls right, rises left
  7. Draw a smooth S-shaped curve through all key points

Deep Dives (click into each)
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🚨 Common Mistakes
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  1. Not rewriting before differentiating: $f(x) = \frac{x^3 + 2x}{x}$ must be simplified to $f(x) = x^2 + 2$ BEFORE using the power rule.
  2. Dropping the limit: In first principles, you must write $\lim_{h \to 0}$ on EVERY line until you actually substitute $h = 0$.
  3. Confusing $f(x)$, $f'(x)$, and $f''(x)$: $f(x)$ gives $y$-values, $f'(x)$ gives gradients, $f''(x)$ gives concavity.
  4. Tangent equation errors: The gradient of the tangent at $x = a$ is $f'(a)$, NOT $f(a)$. Then use $y - f(a) = f'(a)(x - a)$.
  5. Stationary point ≠ inflection point: Stationary points have $f'(x) = 0$. The inflection point has $f''(x) = 0$. Don’t confuse them.
  6. Optimization — not checking for max/min: After finding $f'(x) = 0$, use $f''(x)$ to confirm whether it’s a maximum or minimum. Don’t assume.

🔗 Related topics:

📌 Grade 10/11 foundations:

⏮️ Polynomials | 🏠 Back to Grade 12 | ⏭️ Probability