Derivatives of Products and Quotients

Product Rule

Let Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x) } and $g(x)$ be differentiable functions. Then, the derivative of their product, $f(x)g(x)$ , is

${\frac {d}{dx}}(f(x)g(x))=f'(x)g(x)+f(x)g'(x)$ .

Examples:

$f(x)=6x+1$ , $g(x)=x^{2}$ . $f'(x)=6$ and $g'(x)=2x$ , so ${\frac {d}{dx}}(6x+1)(x^{2})=6x^{2}+(2x)(6x+1)$ .
$f(x)={\sqrt {x}}$ , $g(x)=\sin {x}$ . $f'(x)={\frac {1}{2{\sqrt {x}}}}$ and $g'(x)=\cos {x}$ , so ${\frac {d}{dx}}({\sqrt {x}})(\sin {x})=({\frac {1}{2{\sqrt {x}}}})(\cos {x})+{\sqrt {x}}\sin {x}$ .