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Calculus II: Vector Calculus and Multivariable Analysis / Multivariable Differential Calculus

The Multivariable Chain Rule

Black-Scholes delta changes over time (Γ⋅dS\Gamma \cdot dS in the next period), and a portfolio's exposure to rates depends on how rates affect both the discount factor and the credit spread simultaneously. The multivariable chain rule computes the derivative of a composite function z=f(x(s),y(s))z = f(x(s), y(s)) by tracking how each intermediate variable contributes to the total rate of change. It is the algebraic embodiment of the differential: dz=fx dx+fy dydz = f_x\,dx + f_y\,dy, converted to a rate by dividing by dsds. Implicit differentiation and the second-order Taylor formula — the tools for constraint optimisation and error analysis — follow directly.
Learning objectives. By the end of this note you should be able to:
  • state and apply the chain rule for z=f(x(s),y(s))z = f(x(s), y(s));
  • draw a tree diagram and read off chain-rule formulas for any intermediate-variable structure;
  • apply the chain rule with two independent variables: ∂z/∂s\partial z/\partial s and ∂z/∂t\partial z/\partial t when x=x(s,t)x = x(s,t), y=y(s,t)y = y(s,t);
  • use implicit differentiation to find dy/dxdy/dx when F(x,y)=0F(x,y) = 0;
  • state the second-order Taylor formula in two variables.

In this lesson

    1. The Basic Chain Rule
    1. Two Independent Variables
    1. Functions of Three or More Variables
    1. Implicit Differentiation
    1. Second-Order Taylor Formula

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