The derivative

The derivative of a function f(x) at a point measures how fast f is changing there. Geometrically, it's the slope of the tangent line to the graph at that point. Formally: 

f'(x) = lim[h→0] (f(x+h) − f(x)) / h

If f(x) = x², then f'(x) = 2x. If f(x) = sin(x), then f'(x) = cos(x). If f(x) = eˣ, then f'(x) = eˣ — this exponential function is its own derivative.

The rules

A handful of rules let you differentiate almost any function built from the standard ingredients:

  • Sum rule: (f + g)' = f' + g'
  • Product rule: (fg)' = f'g + fg'
  • Chain rule: (f(g(x)))' = f'(g(x)) · g'(x)
  • Quotient rule: (f/g)' = (f'g − fg') / g²

What it's for

Anywhere one quantity depends on another and we want to understand how they interact:

  • Physics: velocity is the derivative of position; acceleration is the derivative of velocity.
  • Optimization: maxima and minima of a function occur where f'(x) = 0.
  • Machine learning: gradient descent follows derivatives to minimize loss functions.
  • Economics: marginal cost and marginal revenue are derivatives.

The historical surprise

Differential calculus was invented independently by Isaac Newton (around 1666) and Gottfried Wilhelm Leibniz (around 1675). The bitter priority dispute between their followers split Anglo-Continental mathematics for over a century. Modern notation — dy/dx, f'(x) — is Leibniz’s; the theory is jointly theirs.

The other half of calculus is integral calculus, and the Fundamental Theorem of Calculus connects them beautifully: differentiation and integration are inverse operations.