What is covered in Grade 12 Mathematics Calculus?

Limits. Derivative from first principles. Rules of differentiation. Equation of a tangent and normal to a curve.

Is Grade 12 Calculus tested in Paper 1 or Paper 2?

Grade 12 Mathematics Calculus is assessed in Paper 1 of the end-of-year examinations, as per the CAPS Mathematics curriculum guidelines.

When is Calculus taught in Grade 12 Mathematics?

Calculus is taught in Term 2 of Grade 12 according to the CAPS Annual Teaching Plan (ATP) for Mathematics in South Africa.

Is Grade 12 Mathematics Calculus on the CAPS curriculum?

Yes. Calculus is part of the official CAPS Mathematics curriculum for Grade 12, Term 2 in South Africa. The content on MathSciBuddy follows the Annual Teaching Plan (ATP) set by the Department of Basic Education.

Is this Grade 12 Mathematics study material free?

Yes. All chapter notes, definitions, formulas, and worked examples on MathSciBuddy are completely free to read. No account or subscription is needed to access CAPS-aligned study notes.

Grade 12 Mathematics

Grade 12 · Term 2Mathematics

Calculus

We introduce the concept of the derivative from first principles and apply differentiation rules to polynomial functions. We find equations of tangents and normals.

Week 5

6.1 Differentiation — Rules & Tangent Lines

Determine the derivative from first principles: $f'(x)=\lim_{h\to0}\frac{f(x+h)-f(x)}{h}$
Apply differentiation rules: constant, power, sum/difference
Find equations of tangent and normal lines to curves

🌍

Real-World Connection

The speedometer in a car measures the DERIVATIVE of position with respect to time — how quickly position changes each instant. The derivative is the mathematical formalisation of 'instantaneous rate of change', which appears everywhere from physics to economics.

Definition

Derivative (First Principles)

The derivative of $f$ at $x$ is the limiting value of the average rate of change as the interval shrinks to zero.

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

Power Rule

\frac{d}{dx}(x^n) = nx^{n-1}

Works for any real $n$

Constant Rule

\frac{d}{dx}(k) = 0

Derivative of any constant is 0

Sum/Difference Rule

\frac{d}{dx}[f(x)\pm g(x)] = f'(x)\pm g'(x)

Differentiate term by term

Property / Rule

Tangent and Normal Lines

The gradient of the tangent to $y=f(x)$ at $x=a$ is $f'(a)$ . The normal is perpendicular to the tangent, so its gradient is $-\frac{1}{f'(a)}$ .

m_{\text{tangent}}=f'(a);\quad m_{\text{normal}}=-\frac{1}{f'(a)}

🚨 Common Mistake

Before differentiating, SIMPLIFY the expression. $f(x)=\frac{x^3+2x}{x}$ should be simplified to $f(x)=x^2+2$ first. The power rule applies ONLY to individual power terms.

Worked Examples

Worked Example

Derivative from first principles

Problem

f'(x)

Worked Example

Tangent line to a curve

Problem

f(x)=x^3-4x+1

Activity — 8 Questions

CAPS Cognitive Level Distribution

L1 · Knowledge2 Q

L2 · Routine Procedures2 Q

L3 · Complex Procedures2 Q

L4 · Problem Solving2 Q

L1 · Knowledge2 marks

Differentiate

f(x)=5x^3

L1 · Knowledge1 mark

Differentiate

y=7

L2 · Routine Procedures3 marks

Differentiate

g(x)=4x^4-3x^2+2x-7

L2 · Routine Procedures2 marks

Find

f'(x)

f(x)=\sqrt{x}=x^{1/2}

L3 · Complex Procedures3 marks

Find the gradient of the tangent to

y=x^3-2x

at the point where

x=-1

L3 · Complex Procedures4 marks

Find the equation of the normal to

f(x)=x^2+3x

x=1

L4 · Problem Solving4 marks

Find the point on

y=x^2

where the tangent is parallel to

y=4x-1

L4 · Problem Solving5 marks

Find

f'(x)

from first principles for

f(x)=\dfrac{1}{x}

Analytical Geometry