Financial Simulation Modeling in Excel

PARTIAL DERIVATIVES

Introductory calculus covers derivatives and integrals. You may remember that derivatives measure the change of a function relative to a change of its underlying variable, that is, the slope of a function. Partial derivatives, on the other hand, may not be something everyone is familiar with. In a basic sense a partial derivative is a regular derivative but taken within the context of a function with multiple variables. For a simple refresher, let's take the derivative of the following function (equation A.1):

For those who still remember how to take derivatives, the answer to this is trivial. However, we will take the rigorous approach and do this step by step. We start by applying the Chain Rule (equation A.2):

Now, “4”, the second term, is a constant so its derivative is zero. The first term can be quickly evaluated using the Power Rule (equation A.3):

Now imagine that equation A.1 had a second variable, y, associated with it (equation A.4):

With two variables, it would be necessary to measure the change of f(x,y) relative to x while keeping y fixed, and vice ...