From first principles — shrink the chord

\frac{f(x+h) - f(x)}{h} = \frac{(x+h)^2 - x^2}{h}

= \frac{x^2 + 2xh + h^2 - x^2}{h}

= \frac{2xh + h^2}{h} = 2x + h

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

the h disappears — what survives is the tangent's gradient

Differentiation

A curve has a different gradient everywhere.

A straight line has one slope and that's the end of it. On a curve the steepness changes with every step you take — so the gradient isn't a number any more. It's a function.

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Key result

\frac{dy}{dx} = \text{gradient at } x

Each position of the tangent drops one point into the lower panel — its slope, plotted against where you are. Together those points draw a second graph: the derivative.

The power rule

y = x^n \;\Longrightarrow\; \frac{dy}{dx} = n\,x^{\,n-1}

The limit gave $2x$ — and that is exactly the line the tangent traced out for us a moment ago. Do the same work for any power and the pattern is always the same: multiply by the power, then knock it down by one.

The derivative answers two different questions at once. Put a number in and you get the steepness there. Look at its sign and you learn the shape: positive means climbing, negative means falling, and zero means the curve has levelled off.

Chapter 12

That's where the rule comes from.

Negative and fractional powers, the second derivative, and twelve graded exam questions with full worked solutions are waiting.

Read the full chapter › Jump to the questions ›