Q.If $u(x)$ and $v(x)$ are differentiable functions with $u(3)=4,\ u'(3)=2,\ v(3)=5,\ v'(3)=-3$, find the value of $(uv)'(3)$ and $\left(\dfrac{u}{v}\right)'(3)$.

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 …