First Principles Differentiation (AQA A-Level Mathematics): Revision Notes

7.1.2 First Principles Differentiation

The following example shows how to find a formula for the gradient of $y = x^2$.

Summary: The gradient of $F(x)$ is given by the formula

$$\lim_{{h \to 0}} \frac{F(x+h) - F(x)}{h}$$

The limit as $h$ approaches 0.

Gradient of $y = x^4$

$\lim_{{h \to 0}} \frac{(x+h)^4 - x^4}{h} = \lim_{{h \to 0}} \frac{x^4 + 4x^3h + 6x^2h^2 + 4xh^3 + h^4 - x^4}{h}$

$= \lim_{{h \to 0}} (4x^3 + 6x^2h + 4xh^2 + h^3) = :highlight[4x^3]$

(Table showing the relationship between $F(x)$ and its gradient for different powers of $x$ )

• $F(x) = x^1$, Gradient = $1x^0$
• $F(x) = x^2$, Gradient = $2x^1$
• $F(x) = x^3$, Gradient = $3x^2$
• $F(x) = x^4$, Gradient = $4x^3$ $F(x) = x^n, \text{ Gradient } = :highlight[nx^{n-1}]$

The gradient function for $x^n$, $n \in \mathbb{R}$ is given by $nx^{n-1}$.

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Further Differentiation by First Principles

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Notation/Terminology

The technical names for the gradient formula is:

• The differential
• The first differential
• The derivative
• The first derivative
• The gradient function
• The rate of change

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