The Derivative from First Principles

≈ 25 min

The derivative measures how fast a function is changing at a single instant. Start with the average rate of change between two points — the slope of the line joining them:

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

As the gap $h$ shrinks to zero, that joining line becomes the tangent, and the average rate becomes the instantaneous rate. That limit is the derivative:

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

Worked example (first principles). Find $f'(x)$ for $f(x)=x^2$ .

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

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

This is the power rule in miniature: $\dfrac{d}{dx}x^n = nx^{n-1}$ . So $\dfrac{d}{dx}x^2 = 2x$ , $\dfrac{d}{dx}x^3 = 3x^2$ , and a constant differentiates to 0.

The derivative $f'(a)$ tells you:

the value of the function at $x=a$

the slope of the tangent line at $x=a$ — the instantaneous rate of change

the area under the graph up to $x=a$

the average of $f$ over all $x$

Differentiate $f(x)=4x^2-3x+7$ . Give $f'(x)$ .

For $f(x)=x^2$ , find $f'(5)$ .

Use first principles to find $f'(1)$ for $f(x)=3x^2$ .