Determine $f^{(5)}(0)$ without computing any derivatives? Determine $f^{(5)}(0)$ without computing any derivatives? How do i approach a problem like this? I think i use maclaurin/taylor series but i'm not completely sure?
$f(x)=x\ln(1+x)$
$f(x)=x\cos x$

$$f(x) = x \log{(1+x)} = \sum_{k=1}^{\infty} (-1)^{k+1} \frac{x^{k+1}}{k}$$
For any sufficiently differentiable function
$$f(x) = \sum_{k=0}^{\infty} \frac{f^{(k)}(0)}{k!} x^k$$
so that
$$\frac{f^{(5)}(0)}{5!} = -\frac{1}{4} \implies f^{(5)}(0) = -30$$
Do a similar thing for 
$$f(x) = x \cos{x} = \sum_{k=0}^{\infty} (-1)^{k} \frac{x^{2 k+1}}{(2 k)!}$$
