I just finished studying about Taylor expansion from a beautiful video by 3Blue1Brown on youtube. I was trying it out on a few questions until I came across this and I don't understand how to handle Taylor Expansion with multiple variable. Could someone help provide and explanation and the solution?

Q) Compute the first three terms of Taylor Expansion of $f:\mathbb{R}\to\mathbb{R}$ given by

$$f(x) = ye^{x^2}$$

for $y\in\mathbb{R}$ around $x_0=0$. Please include intermediate steps.

  • $\begingroup$ Also the symbol $y$ is used, it's not a variable. Notice the function is defined from $\mathbb R$ to $\mathbb R$. The $y$ is just a constant multiplying $e^{x^2}$. $\endgroup$ – Matthew Leingang Feb 26 at 20:07
  • $\begingroup$ PS Actual Taylor expansion with two variables is a much more involved process. A function $f(x,y)$ has two first derivatives and four second derivatives (although two of them are usually the same). $\endgroup$ – Matthew Leingang Feb 26 at 20:08
  • $\begingroup$ Please use MathJax to format your posts. You will get a much more positive response if you questions are easy to read. $\endgroup$ – saulspatz Feb 26 at 20:13
  • $\begingroup$ Oh yes you are right about y. I'm sorry. $\endgroup$ – Aurangzeb Rathore Feb 26 at 20:25

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