Calculus

What Happens When a Function Has 100 Variables?

Differentiating functions of several variables: partials, the gradient, and directional derivatives.

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What to remember

  • ∂f/∂x\partial f/\partial x differentiates in xx with all other variables frozen.
  • To first order a move splits by input, Δf≈∑i∂f∂xi Δxi\Delta f \approx \sum_i \frac{\partial f}{\partial x_i}\,\Delta x_i; differentiable means the leftover shrinks faster than the move, which is why a risk report adds up the Greeks.
  • Clairaut: continuous mixed partials commute, so Hessians are symmetric.
  • The gradient points toward steepest ascent; directional derivative Duf=∇f⋅uD_u f = \nabla f \cdot u.
Read the lessonPartial Derivatives, with a checkpoint at the end.