Question

Find the derivative of the following function (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):

${\displaystyle \frac{ax+b}{p{x}^2+qx+r}}$

Answer 1

${\displaystyle \frac{d}{dx}\left(\frac{ax+b}{p{x}^2+qr+r}\right)=\frac{\left[\frac{d}{dx}(ax+b)\right](p{x}^2+qx+r)-(ax+b)\frac{d}{dx}(p{x}^2+qx+r)}{{(p{x}^2+qx+r)}^2}}$

= ${\displaystyle \frac{a.(p{x}^2+qx+r)-(ax+b).(2px+q)}{{(p{x}^2+qx+r)}^2}}$

= ${\displaystyle \frac{(ap{x}^2+aqx+ar)-[2ap{x}^2+(aq+2bp)x+bq]}{{(p{x}^2+qx+r)}^2}}$

= ${\displaystyle \frac{-ap{x}^2+ar-2bpx-bq}{{(p{x}^2+qx+r)}^2}}$

= ${\displaystyle \frac{-ap{x}^2-2bpx+ar-bq}{{(p{x}^2+qx+r)}^2}}$