Leaving Cert Calculus (Higher Level): Differentiation & Integration

Calculus on the Leaving Cert Higher course covers differentiation (rates of change, slopes, max/min) and integration (areas, antiderivatives). It appears on Paper 1 and is worth significant marks every year.

Written by Sean Doyle · Last updated: 2026-06-12

Key facts

Calculus is examined on Paper 1 and is one of the most heavily weighted topics on the Higher Level course.
Differentiation finds the slope of a curve at a point; integration reverses it to find area under a curve.
The power rule, product rule, quotient rule and chain rule are the four core differentiation tools.
Most calculus exam questions reward method marks even when the final answer is wrong.

Calculus explained

Understanding Calculus Notation

Calculus uses specific notation to express mathematical ideas. Understanding this notation is essential for working with derivatives and integrals.

Key Notation: - $\frac{dy}{dx}$ or $f'(x)$ represents the derivative of $y$ with respect to $x$ - $\int$ represents integration (finding the area under a curve) - $\lim$ represents a limit (what a function approaches)

Derivative Notation

There are several ways to write derivatives:

Leibniz Notation: $\frac{dy}{dx}$ - reads as "dy by dx"

Prime Notation: $f'(x)$ or $y'$ - reads as "f prime of x"

Newton Notation: $\dot{y}$ - used in physics for time derivatives

All of these mean the same thing: the rate of change of $y$ with respect to $x$ .

Example: Derivative Notation

If $y = x^2 + 3x$ , we can write its derivative as:

$\frac{dy}{dx} = 2x + 3$

$f'(x) = 2x + 3$

$y' = 2x + 3$

All three notations express the same derivative.

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Key formulas

Name	Formula	Description
Power Rule (Differentiation)	$\frac{d}{dx}(x^n) = nx^{n-1}$	Derivative of x to the power n
Product Rule	$\frac{d}{dx}(uv) = u\frac{dv}{dx} + v\frac{du}{dx}$	Derivative of a product of two functions
Quotient Rule	$\frac{d}{dx}\left(\frac{u}{v}\right) = \frac{v\frac{du}{dx} - u\frac{dv}{dx}}{v^2}$	Derivative of a quotient of two functions
Chain Rule	$\frac{dy}{dx} = \frac{dy}{du} \times \frac{du}{dx}$	Derivative of a composite function
Derivative of sin x	$\frac{d}{dx}(\sin x) = \cos x$	Derivative of sine function
Derivative of cos x	$\frac{d}{dx}(\cos x) = -\sin x$	Derivative of cosine function

Worked examples

Worked example 1

For $x^2 + y^2 = 16$ , what is $\frac{dy}{dx}$ ?

Differentiate both sides: 2x + 2y(dy/dx) = 0. Solving for dy/dx: 2y(dy/dx) = -2x, so dy/dx = -2x/(2y) = -x/y

Answer:

-x/y

Worked example 2

Which notation represents the second derivative of $y$ with respect to $x$ ?

The second derivative can be written as d²y/dx² in Leibniz notation or y'' in prime notation. Option B shows the correct Leibniz notation for the second derivative.

Answer:

d²y/dx²

Where students lose marks

Forgetting to apply the chain rule when differentiating composite functions.
Dropping the constant of integration (+C) on indefinite integrals.
Mixing up the derivative and integral of trig functions.
Not setting the derivative equal to zero when finding turning points.

Frequently asked questions

Is calculus hard on the Leaving Cert?

Calculus is very learnable once you know the rules. The same question types repeat every year, so practising past papers makes it one of the most reliable sources of marks.

What calculus do I need for Leaving Cert Higher Maths?

You need differentiation (power, product, quotient and chain rules), applications like rates of change and max/min, and integration including definite integrals and area under a curve.

Is calculus on Paper 1 or Paper 2?

Calculus is on Paper 1 of Leaving Cert Higher Level Maths.

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