Differentiation from first principles

The definition every differentiation rule is built on — the gradient as the limit of a chord — and the "cancel the h, then let h → 0" technique for using it directly.

Pure · Differentiation

Differentiation from first principles

The definition every differentiation rule is built on — the gradient as the limit of a chord — and the "cancel the h, then let h → 0" technique for using it directly.

Why it works

Every differentiation rule you'll ever use rests on a single definition, and "from first principles" means going back to it instead of using a shortcut.

The gradient of a straight line is easy — rise over run. A curve's gradient keeps changing, so we sneak up on it. Take the point at

x

and a second point a small step

h

further along, at

x + h

. The straight line joining them (a chord) has gradient

\frac{f(x+h) - f(x)}{h} \quad (\text{rise over run}).

Now slide that second point in, letting

h

shrink toward

0

: the chord pivots until it becomes the tangent, and its gradient becomes the gradient of the curve itself. That limit is the derivative:

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

Here's the move that makes it actually work. You can't just put

h = 0

straight away — the fraction collapses to

\frac{0}{0}

, which tells you nothing. Instead you simplify first: expand

f(x+h)

, subtract

f(x)

, and you'll find every surviving term on top still carries a factor of

h

. That

h

cancels the one underneath, leaving an expression with no

h

in the denominator. Now it is safe to let

h \to 0

— set the leftover

h

's to zero — and read off the answer.

For

f(x) = x^2

f(x+h) = (x+h)^2 = x^2 + 2xh + h^2

. Subtract

f(x)

2xh + h^2

. Divide by

h

2x + h

. Let

h \to 0

2x

— exactly the power rule, but now you have seen where it comes from. That's the whole point: the shortcut rules are theorems, and this is their proof.