Taylor Polynomial Approximation

[y_lindsay]

How to find a polynomial P(x) of the smallest degree such that sin(x-x^2)=P(x)+o(x) as x->0?
Do I have to calculate the first six derivatives of f(x)=sin(x-x^2) to get Taylor polynomial approximation?

 

[HallsofIvy]

If you know the Taylor series expansion for sin(x) you don't need to calculate any derivatives at all. Just replace x in the expansion with x- x².

$sin(x)= x- x^3/6+ x^5/5!+ \cdot\cdot\cdot$.
$sin(x-x^2)= (x-x^2)- (x-x^2)^3+ (x-x^2)^5/5!+ \cdot\cdot\cdot$