Calculus

Lesson 8 of 14

Partial Derivatives

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

One variable at a time

A portfolio's value depends on many inputs at once: spot, vol, rates, time. How do you measure each effect separately? Hold everything else fixed.

For f(x,y)f(x,y), the partial derivative ∂f∂x\frac{\partial f}{\partial x} differentiates with respect to xx treating yy as a constant. If f(x,y)=x2y+exyf(x,y) = x^2 y + e^{xy}:

fx=2xy+yexy,fy=x2+xexyf_x = 2xy + y e^{xy}, \qquad f_y = x^2 + x e^{xy}