Define Derivative of Product of Polynomials

Question

I have a a problem with defining a certain term...

The derivative of a product of polynomials is the sum of derivatives of the products of the summands of the polynomials of the original product. (right?)

Say we have a product $P$ of n polynomials $p_i$:

$P \equiv \prod_1^n p_i$

Then

$P' = \sum (\square)'$

where $\square$ are the products of $n$ factors of the summands of the polynomials $p_i$.

So for example:

$a \equiv a_1 + a_2$

$b \equiv b_1 + b_2$

Then

$(a \times b)' = ((a_1+a_2)\times(b_1 + b_2))' = (a_1 \times b_1)' + (a_1 \times b_2)' + (a_2 \times b_1)' + (a_2 \times b_2)'$

So, the summands in the last part are all my $\square$s. And they all have 2 factors, because my initial product has two polynomials in it.

My question now is: How do I define those squares aka "the products of $n$ factors of the summands of the polynomials $p_i$"?

I hope I could convey my question properly.

Answer 1

$P' = \sum_i ( p_i' \prod_{j \neq i} p_j )$.

Answer 2

Howabout

$$ P'=p_1\cdots p_n \left( \frac{p'_1}{p_1}+\cdots+\frac{p'_n}{p_n}\right) $$

For a quick derivation, let $P=p_1\cdots p_2$.

Then $$ \ln(P)=\ln(p_1 \cdots p_n)=\ln(p_1)+\cdots+\ln(p_n)$$

Differentiate both sides to get: $$ \frac{P'}{P}=\frac{p_1'}{p_1}+\cdots+\frac{p_n'}{p_n}$$ Hence, $$ P'=P\left( \frac{p_1'}{p_1}+\cdots+\frac{p_n'}{p_n} \right)$$ $$ = p_1\cdots p_n \left( \frac{p_1'}{p_1}+\cdots+\frac{p_n'}{p_n} \right)$$