The Derivative (VCE SSCE Mathematical Methods): Revision Notes

The Derivative

This note introduces the fundamental concept of the derivative, which measures how a function changes at any given point. Understanding derivatives is essential for analysing rates of change in mathematical models and solving real-world problems.

Average rate of change

When working with a function $y = f(x)$, we can calculate how much the function changes on average over a particular interval. This is called the average rate of change.

For any function $y = f(x)$, the average rate of change over the interval $[a, b]$ is found by calculating the gradient (slope) of the line connecting two points on the curve: $A(a, f(a))$ and $B(b, f(b))$.

The formula is:

$$\text{average rate of change} = \frac{f(b) - f(a)}{b - a}$$

This formula calculates how much the $y$-value changes divided by how much the $x$-value changes. Geometrically, this represents the slope of the secant line through the two points.

The tangent to a curve at a point

To understand derivatives, we need to distinguish between different types of lines related to curves:

• A chord is a line segment joining two points on a curve
• A secant is a line passing through two points on a curve
• A tangent is a line that touches the curve at exactly one point

Finding the derivative of $x^2$

Let's develop this idea using the function $f(x) = x^2$.

Consider a point $P(a, a^2)$ and another point $Q(a + h, (a + h)^2)$ on the curve, where $h$ is a small number.

The gradient of the secant line $PQ$ is:

$$\text{gradient of } PQ = \frac{(a + h)^2 - a^2}{a + h - a}$$

Expanding the numerator:

$$$$ $$$$ $$$$

As $h$ gets closer and closer to $0$, the expression $2a + h$ approaches $2a$. We say that the limit of $2a + h$ as $h$ approaches $0$ is $2a$.

Therefore, the gradient of the tangent at point P is 2a.

This result works for any value on the curve, not just at the specific point $a$. The gradient of the tangent to $y = x^2$ at any point $x$ is $2x$.

We say that the derivative of x² with respect to x is 2x, or simply, the derivative of $x^2$ is $2x$.

Limit notation

In mathematics, we use special notation to describe limits. The notation for "the limit of $2x + h$ as $h$ approaches $0$" is:

$$\lim_{h \to 0} (2x + h)$$

This limit equals $2x$ because as $h$ gets very close to $0$, the expression $2x + h$ gets very close to $2x$.

To find the derivative of a function $f(x)$, we follow this process:

1. Find an expression for the gradient of the secant line through points $P(x, f(x))$ and $Q(x + h, f(x + h))$
2. Calculate the limit of this expression as $h$ approaches $0$

Evaluating limits

Definition of the derivative

Now we can state the formal definition of the derivative.

Consider a function $f: \mathbb{R} \to \mathbb{R}$ with graph $y = f(x)$.

The gradient of the secant line $PQ$ connecting points $P(x, f(x))$ and $Q(x + h, f(x + h))$ is:

$$\text{Gradient of secant } PQ = \frac{f(x + h) - f(x)}{x + h - x} = \frac{f(x + h) - f(x)}{h}$$

The gradient of the tangent line at point $P(x, f(x))$ is the limit of this expression as $h$ approaches $0$.

Derivative of a function

The tangent line to the graph of function $f$ at point $(a, f(a))$ is the line passing through $(a, f(a))$ with gradient $f'(a)$.

Differentiation by first principles

Finding the derivative of a function by directly evaluating the limit in the definition is called differentiation by first principles. This method involves algebraic manipulation followed by taking a limit.

Approximating the value of the derivative

From the definition of the derivative, we can see that for a small value of $h$:

$$f'(a) \approx \frac{f(a + h) - f(a)}{h}$$

This formula approximates the derivative by calculating the gradient of the secant line through points $P(a, f(a))$ and $Q(a + h, f(a + h))$. This is called the forward difference approximation.

However, we can often get a better approximation using the central difference approximation:

$$f'(a) \approx \frac{f(a + h) - f(a - h)}{2h}$$

This formula calculates the gradient of the secant line through points $R(a - h, f(a - h))$ and $Q(a + h, f(a + h))$.

These approximation methods are particularly useful when working with functions that are difficult to differentiate analytically, or when estimating derivatives from numerical data.