Product rule

The product rule for derivatives states that

ddx(f(x)+g(x))=ddxf(x)+ddxg(x)=f′(x)+g′(x)\frac{d}{dx}(f(x)+g(x))=\frac{d}{dx}f(x)+ \frac{d}{dx}g(x)=f'(x)+g'(x)

The same applies to partial derivatives. The general case for a product of nn functions is

ddx∏i=1nfi(x)=(∏i=1nfi(x))(∑i=1nfi′(x)fi(x))\frac{d}{dx}\prod_{i=1}^{n}f_{i}(x)=\left( \prod_{i=1}^{n} f_{i}(x) \right)\left( \sum_{i=1}^{n} \frac{f'_{i}(x)}{f_{i}(x)} \right)