Differentiation Revision Notes for AQA GCSE Further Maths

Differentiation

What is differentiation?

Differentiation is a fundamental concept in calculus that helps us find the exact rate of change of a function at any given point. Instead of trying to draw a perfectly accurate tangent line by hand, we use a mathematical approach that starts with easier-to-calculate chord gradients and then applies the concept of limits.

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Think of differentiation as a mathematical magnifying glass - it allows us to zoom in on a curve until we can see the exact slope at any single point, no matter how curved the line appears.

Understanding chords and gradients

When we have a curve, we can draw straight lines connecting any two points on that curve. These connecting lines are called chords or secant lines. The gradient (or slope) of a chord tells us the average rate of change between those two points.

Let's look at the parabola $y = x^2$ . If we take point P at $(3, 9)$ and connect it to various other points Q on the curve, we can calculate the gradient of each chord using the familiar formula:

$\text{Gradient} = \frac{\text{change in y}}{\text{change in x}} = \frac{y\_2 - y\_1}{x\_2 - x\_1}$

Notice something fascinating in this table: as point Q gets closer and closer to point P, the chord gradients are getting closer and closer to the value 6. This suggests that the gradient of the tangent line at $P(3, 9)$ is exactly 6.

The limit approach

To understand this process more generally, let's consider what happens when we make the distance between our two points infinitely small.

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Worked Example: Finding the Tangent Gradient Using Limits

Starting with point P at $(3, 9)$ and point Q at $(3 + h, (3 + h)^2)$ , we can calculate the gradient of chord PQ:

Step 1: Set up the gradient formula $\text{Gradient of PQ} = \frac{(3 + h)^2 - 9}{h}$

Step 2: Expand and simplify

$(3 + h)^2 = 9 + 6h + h^2$
So the gradient becomes: $\frac{9 + 6h + h^2 - 9}{h} = \frac{6h + h^2}{h}$

Step 3: Factor and cancel $\frac{6h + h^2}{h} = \frac{h(6 + h)}{h} = 6 + h$

Step 4: Apply the limit As $h \to 0$ , the gradient approaches $6 + 0 = 6$ .

This limit process gives us the exact gradient of the tangent at point P.

The gradient function

Rather than calculating the gradient at just one specific point, we can develop a general formula that works for any point on the curve.

Worked example: Finding the gradient function for $y = x^2$

For the curve $y = x^2$ , let's take a general point P at $(x, x^2)$ and point Q at $(x + h, (x + h)^2)$ . This approach will give us a formula that works for any x-value.

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Worked Example: Deriving the General Gradient Function

Step 1: Set up the gradient formula for general points $\text{Gradient of chord PQ} = \frac{(x + h)^2 - x^2}{h}$

Step 2: Expand the numerator $(x + h)^2 - x^2 = x^2 + 2xh + h^2 - x^2 = 2xh + h^2$

Step 3: Simplify the fraction $\text{Gradient} = \frac{2xh + h^2}{h} = \frac{h(2x + h)}{h} = 2x + h$

Step 4: Apply the limit as $h \to 0$ The gradient approaches $2x + 0 = 2x$

Result: For the curve $y = x^2$ , the gradient at any point $(x, x^2)$ is $2x$ .

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This means we now have a gradient function! Instead of calculating the gradient at individual points, we have a formula $\text{gradient} = 2x$ that instantly tells us the slope at any x-coordinate on the curve $y = x^2$ .

General method

This same approach works for any polynomial function. The key steps are:

Take two points: P at $(x, y)$ and Q at $(x + h, y + \text{small change})$
Calculate the gradient of the chord PQ
Simplify the expression
Find what happens as $h$ approaches zero

Alternative notation

Mathematicians often use different symbols to represent small changes. Instead of $h$ , we might use $\delta x$ (delta x) to represent a small change in x, and $\delta y$ (delta y) for the corresponding small change in y.

The gradient of a chord becomes $\frac{\delta y}{\delta x}$ .

When we take the limit as $\delta x$ approaches zero, we write this as:

$\frac{dy}{dx} = \lim\_{\delta x \to 0} \frac{\delta y}{\delta x}$

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This notation $\frac{dy}{dx}$ is read as "dy by dx" and represents the derivative of y with respect to x. It's one of the most important pieces of mathematical notation you'll encounter in calculus.

Key differentiation rules

From our work above, we can establish some important patterns that will save you enormous amounts of time:

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The Power Rule for Differentiation

If $y = x^2$ , then $\frac{dy}{dx} = 2x$
If $y = x^3$ , then $\frac{dy}{dx} = 3x^2$
If $y = x^4$ , then $\frac{dy}{dx} = 4x^3$

In general: If $y = x^n$ , then $\frac{dy}{dx} = nx^{n-1}$

This is called the power rule for differentiation and is your most powerful tool for differentiating polynomial functions.

Understanding the terminology

The process we've described is called differentiation. When we differentiate a function, we find its derivative.

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What the derivative $\frac{dy}{dx}$ represents:

The gradient of the tangent to the curve at any point
The instantaneous rate of change of y with respect to x
How quickly y is changing compared to x at that exact moment

Think of it as the "speedometer reading" of how fast your function is changing at any given instant.

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Key Points to Remember:

Chords connect two points on a curve and give us average rates of change
Tangents touch the curve at exactly one point and give us instantaneous rates of change
The limit process allows us to find exact gradients by making chord lengths approach zero
$\frac{dy}{dx}$ notation represents the derivative and means "rate of change of y with respect to x"
The power rule $(y = x^n \Rightarrow \frac{dy}{dx} = nx^{n-1})$ is your most important differentiation tool for polynomial functions

Differentiation (AQA GCSE Further Maths): Revision Notes