An average gradient uses two points: delta ydelta x. As the second point approaches the first, the secant approaches the tangent. Its limiting gradient is dydx. This is an informal limit, not a requirement to differentiate from first principles. The second derivative d2ydx2 measures how the gradient changes. For y=x2, these derivatives are 2x and 2. The blue secant y=3x-2 joins (1,1) to (2,4); the tangent at (1,1) is y=2x-1.

An average gradient uses two points: \(\dfrac{\delta y}{\delta x}\). As the second point approaches the first, the secant approaches the tangent. Its limiting gradient is \(\dfrac{\mathrm{d}y}{\mathrm{d}x}\). This is an informal limit, not a requirement to differentiate from first principles.

The second derivative \(\dfrac{\mathrm{d}^2y}{\mathrm{d}x^2}\) measures how the gradient changes. For \(y=x^2\), these derivatives are \(2x\) and \(2\).

The blue secant \(y=3x-2\) joins \((1,1)\) to \((2,4)\); the tangent at \((1,1)\) is \(y=2x-1\).

From secants to tangents

From secants to tangents