First Principles for Derivatives (HSC SSCE Mathematics Advanced): Revision Notes

First Principles for Derivatives

What is a derivative?

A derivative tells us how quickly a function is changing at any specific point. It represents the instantaneous rate of change of the function with respect to its variable.

Geometrically, the derivative gives us the gradient of the tangent line to the curve at a particular point. The derivative of $y$ with respect to $x$ is written as $\frac{dy}{dx}$ or $y'$.

Key terminology

Derivative

The derivative of a function measures the instantaneous rate of change. It shows us the gradient of the tangent line to the function's graph at any given point.

First principles

First principles is a method for finding the derivative using the limit definition. This fundamental approach forms the basis for all differentiation rules and can be applied to various functions, including linear and quadratic functions.

The first principles formula is:

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

Limit

A limit is the value that a function or sequence approaches as the input approaches some value. In differentiation, the limit describes the value that the gradient of the secant line approaches as the interval between two points approaches zero.

Understanding the geometric interpretation

To understand first principles, we need to think about how a tangent line relates to a secant line.

The derivative of a function $f(x)$ at a point $x$ gives the gradient of the tangent to the curve $y = f(x)$ at that point. We can find this by considering the gradient of a secant line through two points on the curve: $P(x, f(x))$ and $Q(x + h, f(x + h))$.

The gradient of the secant line $PQ$ is:

$m_{\text{Secant}} = \frac{f(x+h) - f(x)}{(x+h) - x}$

$= \frac{f(x+h) - f(x)}{h}$

Where:

• $m_{\text{Secant}}$ is the gradient of the secant line
• $f(x + h) - f(x)$ is the change in the $y$-value (rise)
• $h$ is the change in the $x$-value (run), representing a small increment from $x$

As point $Q$ moves closer to $P$, the value of h approaches zero and the secant line PQ approaches the position of the tangent line at $P$. The gradient of this tangent line is the limiting value of the gradient of the secant line as $h \to 0$.

This limiting value is defined as the derivative of f(x) with respect to x, denoted by $f'(x)$. The process of finding the derivative using this definition is called differentiation from first principles.

The first principles formula

Let's understand each component:

• $f'(x)$ is the derivative of the function $f(x)$
• $\lim_{h \to 0}$ denotes the limit as h approaches zero
• $f(x + h)$ is the value of the function at $x + h$
• $f(x)$ is the value of the function at $x$
• $h$ is a small change in x

This definition is valid provided the limit exists and the tangent is not vertical.

Step-by-step process

To find the derivative of a function using first principles:

1. Find $f(x + h)$: Substitute $x + h$ into the function in place of $x$
2. Expand and simplify: Expand any brackets and simplify the expression
3. Write the first principles formula: Set up $f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}$
4. Substitute: Replace $f(x + h)$ and $f(x)$ with their expressions
5. Simplify the numerator: Expand brackets and collect like terms
6. Factorise: Factor out h from the numerator
7. Cancel the common factor: Cancel $h$ from numerator and denominator
8. Evaluate the limit: Substitute $h = 0$ into the simplified expression

Worked examples

Exam tips

Summary