Everything you need to master Grade 12 Differential Calculus — all in one place
Focus: Rates of change & slopes
Key Concept: Derivatives
Answers: "How fast?" and "How steep?"
Focus: Accumulation & areas
Key Concept: Integrals
Note: Not covered in Grade 12 CAPS
In Grade 12, you'll master:
Limits give us the tools to handle "infinity" and "instantaneous" change mathematically.
Derivatives tell us the rate of change at any point — the "slope" of a curve.
Use derivatives to solve problems: motion, graphs, optimization, and more!
Limits describe the behavior of a function as $x$ approaches a particular value.
The limit of $f(x)$ as $x$ approaches $a$ is $L$:
$$\lim_{x \to a} f(x) = L$$What this means:
As $x$ gets closer and closer to $a$ (from both sides), the function values $f(x)$ get closer and closer to $L$.
The limit as $x$ approaches $a$ from the left (values smaller than $a$).
The limit as $x$ approaches $a$ from the right (values larger than $a$).
The limit exists only if the left-hand and right-hand limits are equal.
Limit of a constant
Limit of $x$
Sum rule
Product rule
The derivative measures the instantaneous rate of change — how fast a function is changing at a specific point.
This is the slope of the secant line between two points on the curve.
This is the slope of the tangent line at a single point. It's what we get by taking the limit as $h$ approaches zero.
Key Insight:
The derivative is the limit of the average rate of change. As the interval gets smaller and smaller ($h \to 0$), we get the instantaneous rate!
The First Principles Formula:
Instead of using first principles every time, we use rules that make differentiation fast and easy.
where $n \in \mathbb{R}$ and $n \ne 0$
Examples:
where $k$ is any constant
Why? A constant doesn't change, so its rate of change is zero!
Examples:
You can "pull out" constants when differentiating.
Examples:
Sum Rule:
$$\frac{d}{dx}[f(x) + g(x)] = f'(x) + g'(x)$$Difference Rule:
$$\frac{d}{dx}[f(x) - g(x)] = f'(x) - g'(x)$$Differentiate each term separately, then add or subtract the results.
Example:
The derivative at a point gives us the gradient of the tangent line at that point.
To find the equation of the tangent line to $f(x)$ at $x = a$:
Step 1: Find the point of tangency
Step 2: Find the gradient
Step 3: Use point-slope form
or rearrange to $y = mx + c$ form
Remember:
The tangent line just touches the curve at one point and has the same slope as the curve at that point.
Calculus is perfect for describing motion because derivatives tell us about rates of change.
Represents: Where the object is
Units: meters (m)
Represents: How fast it's moving
Units: m/s
Represents: How fast velocity changes
Units: m/s²
The Relationships:
Key Insights:
The derivative tells us where a function is increasing, decreasing, and where it has turning points.
When the derivative is positive, the function is increasing (going uphill).
When the derivative is negative, the function is decreasing (going downhill).
When the derivative equals zero, the function has a horizontal tangent — a turning point!
Maximum:
$f'(x)$ changes from + to −
(increasing → decreasing)
Minimum:
$f'(x)$ changes from − to +
(decreasing → increasing)
Find $f'(x)$
Differentiate the function
Set $f'(x) = 0$ and solve
Find the $x$-coordinates of stationary points
Test intervals or use second derivative
Determine if each point is a max, min, or point of inflection
The second derivative is the derivative of the derivative. It tells us about the concavity and helps classify turning points.
The second derivative measures the rate of change of the rate of change. For motion, if $s(t)$ is position, then $s''(t)$ is acceleration!
The curve opens upward like a cup.
The slope is increasing
The curve opens downward like a frown.
The slope is decreasing
If $f'(a) = 0$ (stationary point at $x = a$):
If $f''(a) < 0$
Concave down → Maximum
If $f''(a) > 0$
Concave up → Minimum
If $f''(a) = 0$
Test inconclusive — use interval testing
Use all your calculus knowledge to sketch the complete graph of a function.
Find Intercepts
y-intercept: set $x = 0$; x-intercepts: set $y = 0$ and solve
Find Stationary Points
Set $f'(x) = 0$ and solve for $x$
Classify Stationary Points
Use second derivative test or interval testing
Find Points of Inflection
Set $f''(x) = 0$ (concavity changes)
Determine Behavior at Extremes
What happens as $x \to \infty$ and $x \to -\infty$?
Sketch
Plot all key points and connect smoothly
Pro Tip:
Create a sign chart showing where $f'(x)$ is positive/negative and where $f''(x)$ is positive/negative. This gives you a complete picture of the graph's behavior!
Use calculus to find maximum and minimum values in real-world situations.
Understand the Problem
What are you maximizing or minimizing? What are the constraints?
Draw a Diagram (if applicable)
Visualize the situation and label variables
Write the Objective Function
Express what you want to optimize in terms of ONE variable
Differentiate
Find the derivative of the objective function
Find and Classify Critical Points
Set derivative = 0, solve, then verify it's a max/min
State the Answer with Units
Give the complete answer in the context of the problem
Common Pitfall:
Always check the domain! Physical quantities like length, time, and volume must be positive. Sometimes the maximum or minimum occurs at a boundary, not at a turning point.
Definition of a Limit:
$$\lim_{x \to a} f(x) = L$$Left-hand limit:
$$\lim_{x \to a^-} f(x)$$Right-hand limit:
$$\lim_{x \to a^+} f(x)$$Limit exists if:
$$\lim_{x \to a^-} f(x) = \lim_{x \to a^+} f(x) = L$$Test your understanding across all calculus topics!
Test everything you've learned! Can you get 100%?
You've covered all of Grade 12 Differential Calculus. Keep practicing, reviewing formulas, and working through problems.You've got this! 🎓