Derivative of capital Pi product

Question

I wanted to find the derivative of this function at $x=6$

$$y= \prod_{i=1}^{10} (x-i) = (x-1)(x-2) \cdots (x-10) $$

without expanding all of the brackets, so I used the product rule to find a pattern. However, the resulting sum tells me that the derivative is zero at every whole number which is obviously not true. I've been over my solution and I can't see how I've gone wrong. Please could someone highlight where I went wrong? Thank you in advance.

\begin{align*} \frac{\textit{d}y}{dx} &= (x-2)(x-3) \cdots (x-10) + (x-1) \frac{d}{dx} \biggl((x-2) \cdots (x-10) \biggr) \\ &= \prod_{i=2}^{10} (x-i) + (x-1) \frac{d}{dx} \biggl(\prod_{i=2}^{10} (x-i) \biggr) \\ &= \prod_{i=2}^{10} (x-i) + (x-1)\prod_{i=3}^{10} (x-i) + (x-1)(x-2)\frac{d}{dx} \biggl(\prod_{i=3}^{10} (x-i) \biggr) \\ &= \prod_{i=2}^{10} (x-i) + (x-1)\prod_{i=3}^{10} (x-i) + (x-1)(x-2)\biggl(\prod_{i=4}^{10} (x-i) \biggr) + \cdots \\ &= \frac{y}{x-1} + \frac{y}{x-2} + \frac{y}{x-3}+\cdots + \frac{y}{x-10} \\ &= \sum_{i=1}^{10} \biggl(\frac{y}{x-i}\biggr) \end{align*}

Answer 1

The pattern is simpler to understand than this. In general suppose $f(x) = (x - a) g(x)$ and we want to compute $f'(a)$. (Here $g(x)$ could be a polynomial or more generally any differentiable function.) Then

$$f'(x) = g(x) + (x - a) g'(x)$$

which gives $\boxed{ f'(a) = g(a) }$. This is particularly easy if you notice that applying the definition of the derivative in this case directly gives

$$f'(a) = \lim_{x \to a} \frac{f(x) - f(a)}{x - a} = \lim_{x \to a} g(x) = g(a).$$

Answer 2

A good strategy for derivatives of products is the logarithmic derivative: If $f(x)=\prod f_k(x)$ then $\ln f(x)=\sum \ln f_k(x)$ and by taking derivatives on both sides $$\frac {f’}f =\sum \frac {f_k’}{f_k}$$ Strictly speaking, we need all $f_k(x)>0$, but using the product rule, you can show that $f(x)\ne 0$ is sufficient.

In your example, this takes you very far.

Answer 3

$y' = \sum_\limits{i=1}^{10} \frac {y}{x-i}$ is a fine way to express your derivative for all those values of $x\in\{1,2,3,4,5,6,7,8,9,10\}$ in which case $y'$ is undefined.

However, $\lim_\limits{x\to n} \sum_\limits{i=1}^{10} \frac {y}{x-i}$ with $n$ in the subset of the natural numbers above, does exist, and equals $y'(n)$