Derivative at a point

Rates of change appear everywhere: how fast a reservoir's level is rising, how fast a rocket's speed changes, how quickly a price moves. The derivative measures such a rate.

For a real function f and a point a in its domain, the derivative at a is f′(a) = lim_{h→0} [f(a + h) − f(a)]/h, whenever this limit exists.

Worked: for f(x) = 3x, [3(2 + h) − 6]/h = 3, so f′(2) = 3.

Worked: for f(x) = 2x² + 3x − 5, f′(−1) = −1 and f′(0) = 3, which makes f′(0) + 3f′(−1) = 0.

Worked: the derivative of sin x at 0 is lim (sin h)/h = 1. A constant function such as f(x) = 3 has derivative 0 at every point.

Geometric meaning: mark P at x = a and Q at x = a + h on the curve y = f(x). The difference quotient is the rise over run from P to Q, the slope of chord PQ. As h → 0, Q slides to P, the chord becomes the tangent at P, and f′(a) = tan ψ, the slope of that tangent.