Most people who've studied calculus have learned about Taylor series, and possibly a numerical optimization algorithm called Newton's method. But it's less well known that Taylor series, as well as Newton's method, can be extended to a multivariate setting. In this video, I'll talk about Newton's method for single variable functions first and show how it can be used to do interesting things like computing square roots. Then I show what the multivariate Newton's method looks like. Since Newton's method heavily uses Taylor polynomials, I also talk about how Taylor polynomials can be generalized to multivariate functions as well. Previous video on Gradients, Hessians, and Directional Second Derivatives: Timestamps 0:00 Intro to Taylor Polynomials 1:02 Singlevariate Newton's Method 3:36 Higher Dimensional Linear and Quadratic Polynomials 5:44 Multivariate Taylor Polynomials 12:45 Multivariate Newton's Method