Question

Find the derivative of $\frac{ax+b}{cx+d}$ where $a,b,c,d$ are fixed non-zero constants.

Answer 1

Let $f\left(x\right)=\frac{ax+b}{cx+d}$
Differentiating with respect to $x$
$\Rightarrow f^{\prime }\left(x\right)=\frac{\left[\right(cx+d\left)\frac{d}{dx}\right(ax+b)-(ax+b\left)\frac{d}{dx}\right(cx+d\left)\right]}{(cx+d{)}^2}$
$\Rightarrow f^{\prime }\left(x\right)=\frac{a(cx+d)-c(ax+b)}{(cx+d{)}^2}$
$\Rightarrow f^{\prime }\left(x\right)=\frac{acx+ad-acx-bc}{(cx+d{)}^2}$
$\therefore f^{\prime }\left(x\right)=\frac{ad-bc}{(cx+d{)}^2}$