Q.If $f(0)=6,\ f'(0)=2,\ g(0)=3,\ g'(0)=-1$ (with $g(0)\neq0$), find $\left(\dfrac{f}{g}\right)'(0)$ using the quotient rule.

Exactly as the algebra of limits builds the limit of a

combination of functions from the limits of the simpler functions inside it, the algebra of

derivatives builds the derivative of a sum, difference, product or quotient of two

differentiable functions $u(x)$ and $v(x)$ from their individual derivatives $u'(x)$ and

$v'(x)$ -- derived directly from the first-principles limit definition rather than merely

asserted. The sum/difference rule, $(u\pm v)'=u'\pm v'$, follows immediately from the limit sum

rule; the product rule, $(uv)'=u'v+uv'$, and the quotient rule, $(u/v)'=(u'v-uv')/v^2$ (for

$v\neq0$), both require the technique of subtracting and adding a strategically chosen term …