Partial Derivative of a Product of a Sequence

Question

I think I need to use the chain rule for this? But I'm not sure how to handle the product of a sequence?

$$ y = \prod_{i=1}^{N} (x_i + 5)^3 $$ I'd like to find $ \frac{\partial y}{\partial x_m} $ for some $m$

I know that the derivative of $(x_i + 5)^3$ is $3(x_i + 5)^2$... and I believe that the product of a sequence becomes a summation, but I'm not quite sure how to approach this.

Answer 1

You can use the product rule, considering $y$ to be a function only of $x_i$.

$$\frac{\partial y}{\partial x_i} = \left(\prod_{j\ne i} (x_j+5)^3\right)\cdot 3(x_i+5)^2 + \frac{\partial }{\partial i}\left(\prod_{j\ne i} (x_j+5)^3\right) \cdot (x_i+5)^3.$$

The second term on the RHS is zero. So, the answer is $$\frac{\partial y}{\partial x_i} = 3(x_i+5)^2\left(\prod_{j\ne i} (x_j+5)^3\right).$$

Answer 2

I'm guessing that each $x_i$ has no dependence on the previous $x_i$. Relative to $x_m$, the product excluding the term, $(x_m + 5)^3$ is a constant.

$\frac{\partial y}{\partial x_m} = \frac{\prod_{i=1}^{N} (x_i + 5) ^ 3}{(x_m + 5)^3} * \partial_{x_m} (x_m + 5) ^ 3$.

$\frac{\partial y}{\partial x_m} = \frac{\prod_{i=1}^{N} (x_i + 5) ^ 3}{(x_m + 5)^3} * 3(x_m + 5)^ 2$.