Question

Find the derivative of f (x)=(x^2+1)^3 (x^2+2)^6 using chain rule.

Answer 1

$\bf f(x)=(x^2)^3(x^2+2)^6\impliedby \textit{product rule on the factors} \\\\\\ \cfrac{df}{dx}=[ 3(x^2+1)^2(2x)][(x^2+2)^6]~+~(x^2+1)^3[6(x^2+2)^5(2x)] \\\\\\ \cfrac{df}{dx}=[6x(x^2+1)^2(x^2+2)^6]~+~[12x(x^2+1)^3(x^2+2)^5] \\\\\\ \cfrac{df}{dx}=\stackrel{\textit{common factor}}{6x(x^2+1)^2(x^2+2)^5}[(x^2+2)+2(x^2+1)] \\\\\\ \cfrac{df}{dx}=6x(x^2+1)^2(x^2+2)^5[3x^2+4]$