Differentiation

Q: What is the primary purpose of differentiation when applied to a curve?

Differentiation is used to find the gradient (the slope) of a curve at any specific point.

Q: What are the two common ways to write the notation for a derivative?

The derivative is commonly written as $\frac{dy}{dx}$ (if the equation is $y = ...$) or as $f'(x)$ (if written as a function $f(x)$).

Q: What is the "Power Rule" formula for differentiating a term like $ax^n$?

If $y = ax^n$, then the derivative is $\frac{dy}{dx} = anx^{n-1}$ .

Q: What is the derivative of a constant value and the derivative of $x$?

The derivative of a constant is 0 , and the derivative of $x$ is 1 .

Q: How do you differentiate terms with negative powers, such as $y = \frac{1}{x^n}$?

First, rewrite the term as $y = x^{-n}$, then apply the power rule to get $-nx^{-n-1}$ .

Q: How do you differentiate a surd, such as $y = \sqrt{x}$?

Rewrite the surd as a fractional power, $y = x^{1/2}$, then apply the power rule to get $\frac{1}{2}x^{-1/2}$ .

Q: How do you differentiate an expression containing multiple terms added or subtracted together?

You differentiate each term separately one by one to find the total derivative.

Q: What is the relationship between the gradient of a tangent and the derivative at a point?

The gradient of the tangent at a specific point is exactly equal to the value of $\frac{dy}{dx}$ at that point.

Q: How do you find the gradient of a normal to a curve?

The gradient of the normal is the negative reciprocal of the tangent gradient, calculated as $-\frac{1}{m}$ .

Q: What equation must you solve to find the coordinates of stationary points?

You must set the derivative to zero and solve $\frac{dy}{dx} = 0$ .

Mathematics (9660) · Oxford AQA International AS Level · 6 min read · Updated: 7 Sept 2026

Welcome to the World of Change!

Welcome to one of the most exciting chapters in your AS Level Maths journey: Differentiation. While the name might sound a bit intimidating, it is actually a very simple tool used to measure how things change.

Think about a car driving down a road. Its position is changing every second. Differentiation helps us calculate exactly how fast that position is changing at any single moment. Whether it's the steepness of a mountain or the rate at which a balloon is inflating, differentiation is the math behind it all!

1. What is Differentiation?

At its heart, differentiation is all about finding the gradient (the slope) of a curve. In your earlier studies, you found the gradient of a straight line. But what if the line is curved? The steepness changes at every point!

Differentiation gives us a "gradient formula" (called the derivative) that tells us the slope of a curve at any point we choose.

Important Notations

You will see two main ways to write a derivative. They mean the same thing, so don't let them confuse you:
1. If the equation is written as $y = ...$, the derivative is written as $\frac{dy}{dx}$ (read as "dy by dx").
2. If the equation is written as a function $f(x) = ...$, the derivative is written as $f'(x)$ (read as "f prime of x").

Did you know? The notation $\frac{dy}{dx}$ literally means "a tiny change in $y$ divided by a tiny change in $x$." It’s just like the $\frac{y_2 - y_1}{x_2 - x_1}$ formula you used for straight lines!

2. The "Power Rule": Your Best Friend

For the Oxford AQA P1 syllabus, you primarily need to know how to differentiate polynomials using the Power Rule. This is a simple two-step process.

The Formula:

If $y = ax^n$, then $\frac{dy}{dx} = anx^{n-1}$

Step-by-Step Explanation:

1. Multiply: Take the existing power ($n$) and multiply it by the number in front ($a$).
2. Subtract: Subtract 1 from the original power.

Memory Aid: Just remember "Bring it down, one power down." Multiply by the power, then drop the power by one.

Example 1: Basic Differentiation

Differentiate $y = x^5$.
- The power is 5. Bring it to the front: $5x$.
- Subtract 1 from the power: $5-1 = 4$.
- Result: $\frac{dy}{dx} = 5x^4$

Example 2: With a Coefficient

Differentiate $y = 3x^2$.
- Multiply the power (2) by the number in front (3): $2 \times 3 = 6$.
- Subtract 1 from the power: $2-1 = 1$.
- Result: $\frac{dy}{dx} = 6x^1$ (which is just $6x$).

Quick Review:
- The derivative of a constant (like $y = 5$) is always 0. (Because a flat line has no slope!)
- The derivative of $x$ is always 1.

3. Handling Negative and Fractional Powers

Sometimes the powers aren't nice whole numbers. Don't worry if this seems tricky at first; the rule stays exactly the same!

Negative Powers

If you have $y = \frac{1}{x^2}$, rewrite it first using index laws: $y = x^{-2}$.
- Bring the $-2$ to the front: $-2x$.
- Subtract 1 from the power: $-2 - 1 = -3$.
- Result: $\frac{dy}{dx} = -2x^{-3}$ or $-\frac{2}{x^3}$.

Fractional Powers (Surds)

If you have $y = \sqrt{x}$, rewrite it as $y = x^{1/2}$.
- Bring $1/2$ to the front: $\frac{1}{2}x$.
- Subtract 1: $\frac{1}{2} - 1 = -\frac{1}{2}$.
- Result: $\frac{dy}{dx} = \frac{1}{2}x^{-1/2}$.

Common Mistake: When subtracting 1 from a negative number, students often go the wrong way. Remember: $-3 - 1 = -4$, not $-2$!

4. Differentiation of Sums and Differences

If you have a long expression like $y = 2x^3 - 5x + 7$, you simply differentiate each part one by one.

Example: Differentiate $y = 2x^3 - 5x + 7$.
- $2x^3$ becomes $6x^2$.
- $-5x$ becomes $-5$.
- $7$ becomes $0$.
- Result: $\frac{dy}{dx} = 6x^2 - 5$.

Key Takeaway: Always simplify your expression into separate $x^n$ terms before you start differentiating!

5. Tangents and Normals

Differentiation is the key to finding the equations of specific lines that touch our curves.

The Tangent

A tangent is a straight line that just brushes against the curve at a specific point. It has the same gradient as the curve at that point.
1. Differentiate the equation to get $\frac{dy}{dx}$.
2. Plug in the $x$-value of your point to find the gradient ($m$).
3. Use the straight-line formula $y - y_1 = m(x - x_1)$ to find the equation.

The Normal

A normal is a line perpendicular (at 90 degrees) to the tangent.
- If the tangent gradient is $m$, the normal gradient is $-\frac{1}{m}$ (the negative reciprocal).

Analogy: Imagine you are standing on a curved hill. The ground you are standing on is the tangent. If you held a flagpole perfectly straight up from the ground, that flagpole is the normal.

6. Stationary Points (Maxima and Minima)

A stationary point is a place where the curve is momentarily "flat." This happens at the very top of a hill (maximum) or the very bottom of a valley (minimum).

At these points, the gradient is zero. So, to find them, we set: $$\frac{dy}{dx} = 0$$

How to find and classify Stationary Points:

1. Differentiate the function to find $\frac{dy}{dx}$.
2. Solve $\frac{dy}{dx} = 0$ to find the $x$-coordinates.
3. Find $y$: Plug your $x$ values back into the original equation.
4. Classify: Use the Second Derivative $\frac{d^2y}{dx^2}$.
- Differentiate $\frac{dy}{dx}$ a second time.
- Plug in your $x$ value:
- If $\frac{d^2y}{dx^2} > 0$ (positive), it is a Minimum (looks like a smile $\cup$).
- If $\frac{d^2y}{dx^2} < 0$ (negative), it is a Maximum (looks like a frown $\cap$).

Memory Trick:
- Positive ($+$) people smile $\rightarrow$ Minimum point.
- Negative ($-$) people frown $\rightarrow$ Maximum point.

7. Increasing and Decreasing Functions

Sometimes you just need to know if a graph is going up or down.
- An increasing function is moving upwards. This means its gradient is positive: $\frac{dy}{dx} > 0$.
- A decreasing function is moving downwards. This means its gradient is negative: $\frac{dy}{dx} < 0$.

Final Summary Checklist

- [ ] Can I use the Power Rule $anx^{n-1}$?
- [ ] Do I remember that constants differentiate to 0?
- [ ] Can I rewrite $\sqrt{x}$ and $\frac{1}{x}$ as powers before differentiating?
- [ ] Do I know that $\frac{dy}{dx} = 0$ at a stationary point?
- [ ] Can I use the second derivative to tell a maximum from a minimum?

Great job! Differentiation is a foundational skill. Once you master the "Bring it down, one power down" rule, you've conquered the hardest part. Keep practicing, and it will become second nature!

Quick check

Can you answer these now?

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

What is the primary purpose of differentiation when applied to a curve?

Differentiation is used to find the gradient (the slope) of a curve at any specific point.

What are the two common ways to write the notation for a derivative?

The derivative is commonly written as $\frac{dy}{dx}$ (if the equation is $y = ...$) or as $f'(x)$ (if written as a function $f(x)$).

What is the "Power Rule" formula for differentiating a term like $ax^n$?

If $y = ax^n$, then the derivative is $\frac{dy}{dx} = anx^{n-1}$.

What is the derivative of a constant value and the derivative of $x$?

The derivative of a constant is 0, and the derivative of $x$ is 1.

How do you differentiate terms with negative powers, such as $y = \frac{1}{x^n}$?

First, rewrite the term as $y = x^{-n}$, then apply the power rule to get $-nx^{-n-1}$.

How do you differentiate a surd, such as $y = \sqrt{x}$?

Rewrite the surd as a fractional power, $y = x^{1/2}$, then apply the power rule to get $\frac{1}{2}x^{-1/2}$.

How do you differentiate an expression containing multiple terms added or subtracted together?

You differentiate each term separately one by one to find the total derivative.

What is the relationship between the gradient of a tangent and the derivative at a point?

The gradient of the tangent at a specific point is exactly equal to the value of $\frac{dy}{dx}$ at that point.

How do you find the gradient of a normal to a curve?

The gradient of the normal is the negative reciprocal of the tangent gradient, calculated as $-\frac{1}{m}$.

What equation must you solve to find the coordinates of stationary points?

You must set the derivative to zero and solve $\frac{dy}{dx} = 0$.

How does the second derivative ($\frac{d^2y}{dx^2}$) help classify stationary points?

If $\frac{d^2y}{dx^2} > 0$, it is a minimum point; if $\frac{d^2y}{dx^2} < 0$, it is a maximum point.

How do you determine if a function is increasing or decreasing in a certain interval?

A function is increasing if $\frac{dy}{dx} > 0$ and decreasing if $\frac{dy}{dx} < 0$.

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