Calculus for Statistics If one were to learn calculus solely for the purpose of learning statistics, what should he focus on? If this is a ridiculous question and the honest answer is “All of it,” that is of course an acceptable answer. But if not, what aspects of calculus are particularly salient for learning statistics? Can certain parts be skipped over or should certain parts be emphasized? Welcoming input from anyone conversant in both advanced calculus and advanced statistics. Thanks in advance!
 A: Learn how to differentiate and integrate, and what each of these concepts mean when applied to multi variate functions. I mean Riemann integrals. This alone will get you a long way in statistics. It’s like knowing how to ride a bus and subway in the city. You can get anywhere within the city limits.
I’m assuming you don’t include the probability theory in your ask, because that is a whole different level of mathematics. This would be like having a 4x4 car, so you get to venture outside your town on your own.
A: In addition to the very good answer by Aksakal, a few more suggestions:

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*Yes, differentiation and integration, especially stuff like the chain rule. Pay particular attention to Taylor series, as jros recommends. A low-order approximation is often helpful.


*A rudimentary grasp of linear algebra, especially matrix operations: matrix multiplication, matrix inversion and why doing it directly is usually not a good idea. Yes, this is not calculus per se, but differentiation is really nothing else than linearization, i.e., taking a nonlinear function and approximating it locally by a linear mapping - which is exactly matrix multiplication. Thus, most textbooks on multivariate calculus presuppose this knowledge, or include a quick introduction.
Useful slightly more advanced linear algebra would be eigenvalues/eigenvectors and common matrix decompositions, for PCA and conditioning analysis.


*Mathematical induction comes in handy once in a while when you deal with discrete probability. Again, not calculus as such, but a frequent prerequisite and helpful.


*Perhaps some understanding of dealing with infinite sums, again for discrete probabilities. Happily, our probabilities are nonnegative, so we can usually do dangerous stuff like interchanging summation and differentiation/integration without worrying about absolute convergence. As ColorStatistics notes, this is especially helpful for time series analysis of the ARIMA variety (which in turn I do not find very helpful for practical use, but I don't know whether you are interested in this), and of course Taylor series per above are a special case of power series.


*Per multiple comments: if you want to go deeper into probability theory, then you want to understand measure theory and more advanced integration, e.g., the Lebesgue integral. You will typically find material like this in books on Real Analysis. However, if the stats textbook you want to understand only presupposes "a working knowledge of calculus", then measure theory is definitely not needed. (Fun, though.)
