Product Rule for Derivatives/General Result

Theorem
Let $f_1 \left({x}\right), f_2 \left({x}\right), \ldots, f_n \left({x}\right)$ be real functions differentiable on the open interval $I$.

then:


 * $\forall x \in I: \displaystyle D_x \left({\prod_{i \mathop = 1}^n f_i \left({x}\right)}\right) = \sum_{i \mathop = 1}^n \left({D_x \left({f_i \left({x}\right)}\right) \prod_{j \mathop \ne i} f_j \left({x}\right)}\right)$

Proof
Proof by induction:

For all $n \in \N_{\ge 1}$, let $P \left({n}\right)$ be the proposition:
 * $\displaystyle D_x \left({\prod_{i \mathop = 1}^n f_i \left({x}\right)}\right) = \sum_{i \mathop = 1}^n \left({D_x \left({f_i \left({x}\right)}\right) \prod_{j \mathop \ne i} f_j \left({x}\right)}\right)$

$P(1)$ is true, as this just says:
 * $D_x \left({f_1 \left({x}\right)}\right) = D_x \left({f_1 \left({x}\right)}\right)$

Basis for the Induction
$P(2)$ is the case:
 * $D_x \left({f_1 \left({x}\right) f_2 \left({x}\right)}\right) = D_x \left({f_1 \left({x}\right)}\right) f_2 \left({x}\right) + f_1 \left({x}\right) D_x \left({f_2 \left({x}\right)}\right)$

which has been proved in Product Rule for Derivatives.

This is our basis for the induction.

Induction Hypothesis
Now we need to show that, if $P \left({k}\right)$ is true, where $k \ge 2$, then it logically follows that $P \left({k+1}\right)$ is true.

So this is our induction hypothesis:


 * $\displaystyle D_x \left({\prod_{i \mathop = 1}^k f_i \left({x}\right)}\right) = \sum_{i \mathop = 1}^k \left({D_x \left({f_i \left({x}\right)}\right) \prod_{j \mathop \ne i} f_j \left({x}\right)}\right)$

Then we need to show:


 * $\displaystyle D_x \left({\prod_{i \mathop = 1}^{k+1} f_i \left({x}\right)}\right) = \sum_{i \mathop = 1}^{k+1} \left({D_x \left({f_i \left({x}\right)}\right) \prod_{j \mathop \ne i} f_j \left({x}\right)}\right)$

Induction Step
This is our induction step:

So $P \left({k}\right) \implies P \left({k+1}\right)$ and the result follows by the Principle of Mathematical Induction.

Therefore:
 * $\displaystyle D_x \left({\prod_{i \mathop = 1}^n f_i \left({x}\right)}\right) = \sum_{i \mathop = 1}^n \left({D_x \left({f_i \left({x}\right)}\right) \prod_{j \mathop \ne i} f_j \left({x}\right)}\right)$ for all $n \in \N$

Mnemonic device

 * $\left({f g}\right)' = f g' + f' g$


 * $\left({f g h}\right)' = f' g h + f g' h + f g h'$

and in general, making sure to exhaust all possible combinations, making sure that there are as many summands as there are functions being multiplied.

Also see

 * Derivative of Product of Real Function and Vector-Valued Function
 * Derivative of Vector Cross Product of Vector-Valued Functions
 * Derivative of Dot Product of Vector-Valued Functions