We can use Taylor series to approximate multi-variable functions as polynomial.
Approximate solutions to near
So we get and . This approximation is good up to 5-th order derivatives.
Composition and Substitution
Sometimes it is hard to work with Taylor series of multi-variable functions. Nevertheless, we can simplify the process by using substitution to transform the function into a function of one variable.
Example: Taylor expand about up to terms of order 3
Let , then
And then we can expand the above equation and discard higher order terms