# How do you differentiate f(x) =x(x+3)^3? using the chain rule?

Oct 30, 2015

I found: $f ' \left(x\right) = {\left(x + 3\right)}^{2} \left(4 x + 3\right)$
I would use the Chain and Product Rule to deal with the multiplication between the two functions $x$ and ${\left(x + 3\right)}^{3}$;
I would then use the Chain Rule (in red) to deal with ${\left(x + 3\right)}^{3}$ deriving first the cube leaving the argument as it is and multiplying the derivative of the argument inside the bracket:
$f ' \left(x\right) = 1 \cdot {\left(x + 3\right)}^{3} + x \cdot \textcolor{red}{3 {\left(x + 3\right)}^{2} \cdot 1} =$
$= {\left(x + 3\right)}^{2} \left[\left(x + 3\right) + 3 x\right] =$
$= {\left(x + 3\right)}^{2} \left(4 x + 3\right)$