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

Differentiability, the Total Differential, and the Jacobian Matrix

Partial derivatives measure how ff changes one variable at a time. But a portfolio's value can change when spot, vol, and rates all move simultaneously. The concept that captures this is differentiability: ff is differentiable at (x0,y0)(x_0,y_0) if its best local approximation is a linear map. That linear map is encoded in the total differential df=fx dx+fy dydf = f_x\,dx + f_y\,dy — the first-order P&L attribution formula. For a vector-valued map F:Rm→Rn\mathbf{F}: \mathbb{R}^m \to \mathbb{R}^n, the total derivative is the Jacobian matrix J=[∂Fi/∂xj]J = [\partial F_i/\partial x_j]: the change-of-basis matrix for density transformations, the sensitivity matrix in multi-asset hedging, and the foundation of the multivariable chain rule.
Learning objectives. By the end of this note you should be able to:
  • state the definition of differentiability at a point in R2\mathbb{R}^2;
  • prove that differentiability implies continuity but not vice versa;
  • state the sufficient condition for differentiability (continuous partial derivatives);
  • compute the total differential df=fx dx+fy dydf = f_x\,dx + f_y\,dy and use it for linear approximation;
  • write the Jacobian matrix J=[∂Fi/∂xj]J = [\partial F_i/\partial x_j] for a vector-valued map;
  • interpret ΔF≈J Δx\Delta\mathbf{F} \approx J\,\Delta\mathbf{x} as the linear approximation to the map.

In this lesson

    1. Differentiability
    1. The Total Differential
    1. The Jacobian Matrix
    1. Summary: Relationships

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