It is given the geometric mean:
$$f(x) =(\prod_{i=1}^{k}x_i)^{1/k}$$ on $R_{++}$
The first derivation is a chain rule: $f'(x)=\frac{1}{k}(\prod_{i=1}^{k}x_i)^{\frac{1}{k}-1} \cdot (\text{inner derivation})$
How do I do the inner derivation: $\frac{d}{dx}\left(\prod_{i=1}^{k}x_i\right)$?
It must be a multivariable ($k$-variable) function: $$f(x)=f(x_1,x_2,\cdots,x_k)=\left(\prod_{i=1}^k x_i \right)^{1/k}.$$ You can take a partial derivative: $$\dfrac{\partial f(x)}{\partial x_{i}}=\frac{1}{k}\cdot x_i^{\frac{1}{k}-1} \left( \prod_{j=1, j\ne i}^k x_j \right)^{1/k}.$$
For example: $f(x_1,x_2,x_3)=(x_1x_2x_3)^{1/3}:$ $$f_{x_1}=\frac13 x_1^{1/3-1}\cdot x_2^{1/3}x_3^{1/3}$$