First Principles

Level 1 - Math I (Physics) topic page in Differentiation.

The derivative can be defined from first principles using the limit definition.

Definition

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

This is also called the difference quotient approach.

Worked Example

Find \(f'(x)\) if \(f(x) = x^2\):

Step 1

\[f'(x) = \lim_{h \to 0} \frac{(x+h)^2 - x^2}{h}\]

Step 2

\[= \lim_{h \to 0} \frac{x^2 + 2xh + h^2 - x^2}{h}\]

Step 3

\[= \lim_{h \to 0} \frac{2xh + h^2}{h}\]

Step 4

\[= \lim_{h \to 0} (2x + h) = 2x\]

Alt Form

\[f'(a) = \lim_{x \to a} \frac{f(x) - f(a)}{x - a}\]