Question

Find the derivative of $(ax+b)(cx+d{)}^2$ where $a,b,c,d$ are fixed non-zero constants.

Answer 1

Let $f\left(x\right)=(ax+b)(cx+d{)}^2$
$\Rightarrow f\left(x\right)=(ax+b)(c^2{x}^2+2cdx+d^2)$
Differentiating with respect to $x$.
$\Rightarrow f^{\prime }\left(x\right)=(c^2{x}^2+2cdx+d^2)\frac{d}{dx}(ax+b)+(ax+b)\frac{d}{dx}(c^2{x}^2+2cdx+d^2)$
$\Rightarrow f^{\prime }\left(x\right)=(c^2{x}^2+2cdx+d^2)\times a+(ax+b)\times (2c^{2}x+2cd)$
$\Rightarrow f^{\prime }\left(x\right)=a(cx+d{)}^2+(ax+b)\times 2c(cx+d)$
$\therefore f^{\prime }\left(x\right)=a(cx+d{)}^2+2c(cx+d\left)\right(ax+b)$