Differentiation

Q: What does the derivative of a function represent graphically?

The derivative represents the gradient (slope) of the curve at a specific point.

Q: What are the two common ways to write the derivative of a function?

The derivative of \(y\) with respect to \(x\) is written as \(\frac{dy}{dx}\) or \(f'(x)\) .

Q: How do you differentiate a term with a constant multiple, such as \(y = ax^n\)?

Keep the constant and multiply it by the derivative of the variable: \(\frac{dy}{dx} = anx^{n-1}\).

Q: What is the rule for differentiating a sum or difference of multiple terms?

Differentiate each term in the expression separately.

Q: What is the derivative of a constant number?

The derivative of any constant (a plain number without a variable) is always 0 .

Q: How should roots and fractions be prepared before differentiation?

Convert them into index form (powers). For example, change \(\sqrt{x}\) to \(x^{1/2}\) and \(\frac{1}{x^n}\) to \(x^{-n}\).

Q: How do you find the gradient of a tangent to a curve at a specific point?

Differentiate the function to find \(\frac{dy}{dx}\), then substitute the \(x\)-coordinate of the point into the derivative.

Q: What is the relationship between the gradient of a tangent and the gradient of a normal?

The gradient of the normal is the negative reciprocal of the tangent's gradient: \(-\frac{1}{m}\).

Q: What is a stationary point and how do you find one?

A stationary point is where the gradient is zero (\(\frac{dy}{dx} = 0\)).

Mathematics 7356 · AQA AS Level · 4 min read · Updated: Sep 7, 2026

Welcome to the World of Differentiation!

Welcome! Today we are diving into Differentiation, a core part of your AS Level Mathematics journey. Don't let the name scare you—differentiation is simply a fancy way of measuring how things change. Whether it's a car accelerating, a population growing, or a ball flying through the air, differentiation helps us find the exact "rate of change" at any single moment. Let's break it down together!

1. What is a Derivative?

At its heart, the derivative is the gradient (slope) of a curve. In GCSE, you found the gradient of a straight line using "rise over run." But what if the line is curved? The gradient changes at every point!

The Concept: Imagine you are zooming in on a curve so much that it looks like a straight line. The gradient of that tiny straight section is the derivative at that point. We write the derivative of \(y\) with respect to \(x\) as \(\frac{dy}{dx}\) or \(f'(x)\).

Analogy: Think of a roller coaster. The derivative tells you exactly how steep the track is at the very spot your carriage is sitting right now.

Differentiation from First Principles

Before we use shortcuts, we need to understand where they come from. This is called "First Principles." We take two points on a curve that are very close together, separated by a tiny distance \(h\).

The formula is:
\(f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}\)

Step-by-Step for \(f(x) = x^2\):
1. Find \(f(x+h)\): \((x+h)^2 = x^2 + 2xh + h^2\)
2. Subtract \(f(x)\): \(x^2 + 2xh + h^2 - x^2 = 2xh + h^2\)
3. Divide by \(h\): \(\frac{2xh + h^2}{h} = 2x + h\)
4. As \(h\) gets closer and closer to zero (\(h \to 0\)), we are left with \(2x\).

Quick Review: The derivative of \(x^2\) is \(2x\). This tells us that for any point on the curve \(y=x^2\), the gradient is always double the x-value!

2. The "Power Rule": Our Best Friend

Don't worry if First Principles feels like a lot of algebra—most of the time, we use a much faster rule! For any term where \(y = x^n\):

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

Memory Aid: "Bring the power down to the front, then subtract one from the power."

Examples:
If \(y = x^5\), then \(\frac{dy}{dx} = 5x^4\)
If \(y = x^{-3}\), then \(\frac{dy}{dx} = -3x^{-4}\) (Be careful with negatives!)
If \(y = x^{1/2}\), then \(\frac{dy}{dx} = \frac{1}{2}x^{-1/2}\)

Important Rules for Combinations:

Constant Multiples: If there is a number in front, just multiply it!
If \(y = 4x^3\), then \(\frac{dy}{dx} = 4 \times 3x^2 = 12x^2\).
Sums and Differences: Just differentiate each part separately.
If \(y = x^2 + 5x\), then \(\frac{dy}{dx} = 2x + 5\).
Constants: The derivative of a plain number (like 7) is always 0 because a flat line has no gradient!

Common Mistake to Avoid: Always rewrite roots or fractions as powers before you start. For example, change \(\frac{1}{x^2}\) to \(x^{-2}\) and \(\sqrt{x}\) to \(x^{1/2}\) first!

3. Tangents and Normals

We can use our derivative to find the equation of specific lines that touch our curves.

Tangent: A straight line that just touches the curve at a point. It has the same gradient as the curve at that point.
Normal: A straight line perpendicular (at 90 degrees) to the tangent. Its gradient is \(-\frac{1}{m}\) where \(m\) is the tangent's gradient.

How to find them:
1. Differentiate to find \(\frac{dy}{dx}\).
2. Plug in your \(x\) value to get the gradient (\(m\)).
3. Use the line equation formula: \(y - y_1 = m(x - x_1)\).

Key Takeaway: The derivative gives you the slope. Once you have the slope and a point, you can find the equation of any line passing through it.

4. Stationary Points: Hills and Valleys

A stationary point is a place on the graph where the gradient is exactly zero (\(\frac{dy}{dx} = 0\)). This is where the graph is momentarily flat.

Types of Stationary Points:
1. Local Maximum: The top of a hill.
2. Local Minimum: The bottom of a valley.

Did you know? Engineers use stationary points to find the most efficient shapes for bridges or to minimize the amount of material needed for a soda can!

The Second Derivative: \(\frac{d^2y}{dx^2}\)

How do we know if a point is a maximum or a minimum without looking at the graph? We differentiate again! This is called the second derivative.

If \(\frac{d^2y}{dx^2} > 0\) (positive), it is a Minimum. (Think: Positive people smile, and a smile looks like a valley!)
If \(\frac{d^2y}{dx^2} < 0\) (negative), it is a Maximum. (Think: Negative people frown, and a frown looks like a hill!)

5. Increasing and Decreasing Functions

Sometimes you just need to know if a graph is going up or down in a certain region.

Increasing: The gradient is positive (\(\frac{dy}{dx} > 0\)).
Decreasing: The gradient is negative (\(\frac{dy}{dx} < 0\)).

Summary Table for Quick Review:

To find Gradient: Find \(\frac{dy}{dx}\)
To find Stationary Points: Set \(\frac{dy}{dx} = 0\)
To find Nature (Max/Min): Check the sign of \(\frac{d^2y}{dx^2}\)
To find Normal Gradient: Use \(-\frac{1}{\text{gradient}}\)

Don't worry if this seems tricky at first! Differentiation is a skill that gets much easier with practice. Just remember: it's all about the slope!

Quick check

Can you answer these now?

Open each question to check the key ideas from this chapter.

What does the derivative of a function represent graphically?

The derivative represents the gradient (slope) of the curve at a specific point.

What are the two common ways to write the derivative of a function?

The derivative of \(y\) with respect to \(x\) is written as \(\frac{dy}{dx}\) or \(f'(x)\).

What is the formula for differentiation from first principles?

\(f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}\)

What is the "Power Rule" for differentiating terms like \(x^n\)?

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

How do you differentiate a term with a constant multiple, such as \(y = ax^n\)?

Keep the constant and multiply it by the derivative of the variable: \(\frac{dy}{dx} = anx^{n-1}\).

What is the rule for differentiating a sum or difference of multiple terms?

Differentiate each term in the expression separately.

What is the derivative of a constant number?

The derivative of any constant (a plain number without a variable) is always 0.

How should roots and fractions be prepared before differentiation?

Convert them into index form (powers). For example, change \(\sqrt{x}\) to \(x^{1/2}\) and \(\frac{1}{x^n}\) to \(x^{-n}\).

How do you find the gradient of a tangent to a curve at a specific point?

Differentiate the function to find \(\frac{dy}{dx}\), then substitute the \(x\)-coordinate of the point into the derivative.

What is the relationship between the gradient of a tangent and the gradient of a normal?

The gradient of the normal is the negative reciprocal of the tangent's gradient: \(-\frac{1}{m}\).

What is a stationary point and how do you find one?

A stationary point is where the gradient is zero (\(\frac{dy}{dx} = 0\)).

How does the second derivative identify a local minimum?

If \(\frac{d^2y}{dx^2} > 0\) (positive) at a stationary point, it is a local minimum.

How does the second derivative identify a local maximum?

If \(\frac{d^2y}{dx^2} < 0\) (negative) at a stationary point, it is a local maximum.

When is a function described as "increasing"?

A function is increasing when its derivative is positive: \(\frac{dy}{dx} > 0\).

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