Take a function that is the product of degree $k$, such as $$f_k (x)=\prod_{n=2}^{k+1} g\left(\frac xn \right), k\ge 1.$$ What methods would on use to find $f^\prime_k(x)$ with respect to $x$ in a closed form?
Using the product rule, it's fairly simple to find the derivative in a recursive form, i.e. $$f^\prime_k(x)=f^\prime_{k-1}(x)\cdot g\left(\frac x{k+1} \right)+f_{k-1}(x)\cdot g^\prime\left(\frac x{k+1} \right);$$ $$f^\prime_1(x)=g^\prime\left(\frac x2 \right),$$ but this seems rather clunky and is rather inefficient for larger values of $k$.
I am stuck in finding a closed form of $f^\prime_k(x)$ , which would be much more elegant for applications such as Taylor's Series. I'm sure it's possible, but could somebody point me in the right direction, as far as derivation methods go?
Logarithmic differentiation is your friend.
$f_k (x)=\prod_{n=2}^{k+1} g(x/n) $.
$\ln f_k (x)=\sum_{n=2}^{k+1} \ln g(x/n) $.
Differentiating, $\dfrac{f_k' (x)}{f_k (x)}=\sum_{n=2}^{k+1} \dfrac{g'(x/n)}{ng(x/n)} $, so $f_k' (x)=f_k (x)\left(\sum_{n=2}^{k+1} \dfrac{g'(x/n)}{ng(x/n)}\right) $.
You can bring $f_k$ inside the sum, but I wouldn't bother unless $k$ is small.