Question

Find the derivative of the following functions(it is to be understood that $a,b,c,d,p,q,r$ and $s$ are fixed non-zero constants and $m$ and $n$ are integers) : $\frac{ax+b}{cx+d}$

Answer 1

Let $f\left(x\right)=\frac{ax+b}{cx+d}$
Thus by quotient rule,
$f^{\prime }\left(x\right)=\frac{(cx+d)\frac{d}{dx}(ax+b)-(ax+b)\frac{d}{dx}(cx+d)}{{(cx+d)}^2}$
$=\frac{(cx+d)\left(a\right)-(ax+b)\left(c\right)}{{(cx+d)}^2}$
$=\frac{acx+ad-acx-bc}{{(cx+d)}^2}=\frac{ad-bc}{{(cx+d)}^2}$