Answer
$-35sinx-xcosx$
Work Step by Step
Trig functions are cyclical in nature, so we need to calculate the first few derivatives before we can see the pattern. We do this using the trig derivatives we know and the product rule. We see that $$f(x)=xsinx$$ $$f'(x)=sinx+xcosx$$ $$f''(x)=2cosx-xsinx$$ $$f'''(x)=-3sinx-xcosx$$ $$f^{(4)}(x)=-4cosx+xsinx$$ $$f^{(5)}(x)=5sinx+xcosx$$
By now the pattern is obvious. The coefficient of the first term is the order of the derivative. The signs of each term change every 4 derivatives, and the placement of sine and cosine alternates. Using this knowledge, we can deduce that $f^{(35)}(X)=-35sinx-xcosx$
