Finding the "order of a root" at a value

Question

I am given the following problem:

The order $p$ of a root $x$ of a function $f$ is the order of the smallest non vanishing derivative at the point $x$. For example, if $f(x) = x^2$, then the order of the root at $x = 0$ is 2. Find the order of the root at $x = 0$ of the function \begin{equation} f(x) = e^{sin(x)}-sin(e^x-1)-1 \end{equation}

I have the following questions regarding the above.

1. What exactly is the meaning of "an order of a root". Is it the same meaning as "multiplicity of a root"? That is, an order of a root of 2 means that there are two roots with the same value as in the example given.
2. In very simple terms, given the above, how do I find the order of the root of the equation given? (Note that the question is asked in the context of using a computer algebra system, so its not required that it be calculated by hand).

Answer 1

Yes, what you should be computing is the multiplicity of the root. In this case the answer is $4$, because the Taylor series of your function centered at $0$ is$$\frac{x^4}{12}+\frac{x^5}8+\cdots$$